Properties

Label 930.2.o.e
Level $930$
Weight $2$
Character orbit 930.o
Analytic conductor $7.426$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40q + 6q^{3} - 40q^{4} - 12q^{7} - 2q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 6q^{3} - 40q^{4} - 12q^{7} - 2q^{9} + 20q^{10} - 6q^{12} - 12q^{13} + 40q^{16} - 12q^{18} - 12q^{19} + 12q^{21} - 24q^{22} + 20q^{25} + 12q^{28} + 8q^{31} + 52q^{33} + 24q^{34} + 2q^{36} + 60q^{37} - 8q^{39} - 20q^{40} + 12q^{42} + 24q^{43} - 12q^{45} + 6q^{48} - 4q^{49} + 14q^{51} + 12q^{52} + 24q^{55} - 12q^{57} - 40q^{64} + 8q^{66} + 64q^{67} - 26q^{69} - 24q^{70} + 12q^{72} + 6q^{75} + 12q^{76} - 68q^{78} - 48q^{79} + 2q^{81} + 4q^{82} - 12q^{84} + 36q^{87} + 24q^{88} + 2q^{90} - 22q^{93} - 40q^{94} + 8q^{97} - 90q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 1.00000i −1.67328 0.447368i −1.00000 0.866025 + 0.500000i −0.447368 + 1.67328i −1.32225 2.29021i 1.00000i 2.59972 + 1.49714i 0.500000 0.866025i
161.2 1.00000i −1.14810 + 1.29687i −1.00000 0.866025 + 0.500000i 1.29687 + 1.14810i 0.742621 + 1.28626i 1.00000i −0.363755 2.97787i 0.500000 0.866025i
161.3 1.00000i −0.556373 + 1.64026i −1.00000 0.866025 + 0.500000i 1.64026 + 0.556373i −1.24695 2.15979i 1.00000i −2.38090 1.82519i 0.500000 0.866025i
161.4 1.00000i −0.474351 1.66583i −1.00000 0.866025 + 0.500000i −1.66583 + 0.474351i 1.75266 + 3.03570i 1.00000i −2.54998 + 1.58038i 0.500000 0.866025i
161.5 1.00000i −0.151443 1.72542i −1.00000 0.866025 + 0.500000i −1.72542 + 0.151443i −1.64767 2.85385i 1.00000i −2.95413 + 0.522605i 0.500000 0.866025i
161.6 1.00000i 0.433517 + 1.67692i −1.00000 0.866025 + 0.500000i 1.67692 0.433517i −1.74230 3.01775i 1.00000i −2.62413 + 1.45395i 0.500000 0.866025i
161.7 1.00000i 0.500302 1.65822i −1.00000 0.866025 + 0.500000i −1.65822 0.500302i −1.28701 2.22916i 1.00000i −2.49940 1.65922i 0.500000 0.866025i
161.8 1.00000i 1.29636 + 1.14867i −1.00000 0.866025 + 0.500000i 1.14867 1.29636i 1.71736 + 2.97455i 1.00000i 0.361105 + 2.97819i 0.500000 0.866025i
161.9 1.00000i 1.62650 0.595404i −1.00000 0.866025 + 0.500000i −0.595404 1.62650i −0.581871 1.00783i 1.00000i 2.29099 1.93685i 0.500000 0.866025i
161.10 1.00000i 1.64686 0.536508i −1.00000 0.866025 + 0.500000i −0.536508 1.64686i 0.615418 + 1.06593i 1.00000i 2.42432 1.76711i 0.500000 0.866025i
161.11 1.00000i −1.69869 0.338296i −1.00000 −0.866025 0.500000i 0.338296 1.69869i −1.24695 2.15979i 1.00000i 2.77111 + 1.14932i 0.500000 0.866025i
161.12 1.00000i −1.69717 + 0.345843i −1.00000 −0.866025 0.500000i −0.345843 1.69717i 0.742621 + 1.28626i 1.00000i 2.76078 1.17391i 0.500000 0.866025i
161.13 1.00000i −1.23550 1.21390i −1.00000 −0.866025 0.500000i 1.21390 1.23550i −1.74230 3.01775i 1.00000i 0.0529076 + 2.99953i 0.500000 0.866025i
161.14 1.00000i −0.449207 + 1.67279i −1.00000 −0.866025 0.500000i −1.67279 0.449207i −1.32225 2.29021i 1.00000i −2.59643 1.50285i 0.500000 0.866025i
161.15 1.00000i −0.346599 1.69702i −1.00000 −0.866025 0.500000i 1.69702 0.346599i 1.71736 + 2.97455i 1.00000i −2.75974 + 1.17637i 0.500000 0.866025i
161.16 1.00000i 1.20548 + 1.24372i −1.00000 −0.866025 0.500000i −1.24372 + 1.20548i 1.75266 + 3.03570i 1.00000i −0.0936563 + 2.99854i 0.500000 0.866025i
161.17 1.00000i 1.28806 1.15797i −1.00000 −0.866025 0.500000i 1.15797 + 1.28806i 0.615418 + 1.06593i 1.00000i 0.318204 2.98308i 0.500000 0.866025i
161.18 1.00000i 1.32888 1.11089i −1.00000 −0.866025 0.500000i 1.11089 + 1.32888i −0.581871 1.00783i 1.00000i 0.531863 2.95248i 0.500000 0.866025i
161.19 1.00000i 1.41853 + 0.993862i −1.00000 −0.866025 0.500000i −0.993862 + 1.41853i −1.64767 2.85385i 1.00000i 1.02448 + 2.81965i 0.500000 0.866025i
161.20 1.00000i 1.68621 + 0.395836i −1.00000 −0.866025 0.500000i −0.395836 + 1.68621i −1.28701 2.22916i 1.00000i 2.68663 + 1.33493i 0.500000 0.866025i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 491.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.e odd 6 1 inner
93.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.o.e 40
3.b odd 2 1 inner 930.2.o.e 40
31.e odd 6 1 inner 930.2.o.e 40
93.g even 6 1 inner 930.2.o.e 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.o.e 40 1.a even 1 1 trivial
930.2.o.e 40 3.b odd 2 1 inner
930.2.o.e 40 31.e odd 6 1 inner
930.2.o.e 40 93.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):

\(T_{7}^{20} + \cdots\)
\(15\!\cdots\!68\)\( T_{11}^{22} + \)\(65\!\cdots\!05\)\( T_{11}^{20} + \)\(14\!\cdots\!04\)\( T_{11}^{18} + \)\(23\!\cdots\!21\)\( T_{11}^{16} + \)\(26\!\cdots\!74\)\( T_{11}^{14} + \)\(21\!\cdots\!62\)\( T_{11}^{12} + \)\(13\!\cdots\!26\)\( T_{11}^{10} + \)\(59\!\cdots\!06\)\( T_{11}^{8} + \)\(17\!\cdots\!44\)\( T_{11}^{6} + 332005777128 T_{11}^{4} + 10332663156 T_{11}^{2} + 276922881 \)">\(T_{11}^{40} + \cdots\)