Properties

Label 624.2.cn.f
Level $624$
Weight $2$
Character orbit 624.cn
Analytic conductor $4.983$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.cn (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56q - 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 4q^{7} + 8q^{13} + 8q^{15} - 4q^{19} + 16q^{21} - 24q^{27} + 36q^{31} + 28q^{33} + 20q^{37} - 16q^{39} + 84q^{43} + 12q^{45} - 12q^{49} + 24q^{55} - 36q^{57} - 24q^{61} + 12q^{63} + 32q^{67} - 36q^{69} - 20q^{73} + 60q^{75} + 32q^{79} - 88q^{85} + 16q^{87} - 28q^{91} - 88q^{93} - 36q^{97} - 44q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1 0 −1.68186 0.413931i 0 −1.44076 1.44076i 0 −0.918532 + 3.42801i 0 2.65732 + 1.39235i 0
305.2 0 −1.55432 0.764251i 0 −0.504580 0.504580i 0 0.836809 3.12301i 0 1.83184 + 2.37579i 0
305.3 0 −1.52267 + 0.825508i 0 1.33870 + 1.33870i 0 −0.566234 + 2.11322i 0 1.63707 2.51396i 0
305.4 0 −1.31722 + 1.12469i 0 1.72616 + 1.72616i 0 0.574711 2.14485i 0 0.470140 2.96293i 0
305.5 0 −1.19063 1.25794i 0 2.36656 + 2.36656i 0 0.332282 1.24009i 0 −0.164807 + 2.99547i 0
305.6 0 −0.315401 + 1.70309i 0 −1.72616 1.72616i 0 0.574711 2.14485i 0 −2.80104 1.07431i 0
305.7 0 0.0464265 + 1.73143i 0 −1.33870 1.33870i 0 −0.566234 + 2.11322i 0 −2.99569 + 0.160768i 0
305.8 0 0.121290 1.72780i 0 −2.19967 2.19967i 0 −1.17763 + 4.39498i 0 −2.97058 0.419131i 0
305.9 0 0.386983 1.68827i 0 1.56802 + 1.56802i 0 0.418596 1.56222i 0 −2.70049 1.30666i 0
305.10 0 1.19941 + 1.24957i 0 1.44076 + 1.44076i 0 −0.918532 + 3.42801i 0 −0.122851 + 2.99748i 0
305.11 0 1.26859 1.17927i 0 −1.56802 1.56802i 0 0.418596 1.56222i 0 0.218643 2.99202i 0
305.12 0 1.43567 0.968940i 0 2.19967 + 2.19967i 0 −1.17763 + 4.39498i 0 1.12231 2.78216i 0
305.13 0 1.43902 + 0.963958i 0 0.504580 + 0.504580i 0 0.836809 3.12301i 0 1.14157 + 2.77431i 0
305.14 0 1.68472 + 0.402146i 0 −2.36656 2.36656i 0 0.332282 1.24009i 0 2.67656 + 1.35501i 0
353.1 0 −1.72861 0.109042i 0 −0.190181 0.190181i 0 −3.71203 + 0.994635i 0 2.97622 + 0.376984i 0
353.2 0 −1.55814 0.756439i 0 −3.14772 3.14772i 0 −2.26593 + 0.607154i 0 1.85560 + 2.35728i 0
353.3 0 −1.51931 0.831691i 0 0.741654 + 0.741654i 0 4.39168 1.17675i 0 1.61658 + 2.52719i 0
353.4 0 −1.49406 + 0.876234i 0 1.23702 + 1.23702i 0 −0.738861 + 0.197977i 0 1.46443 2.61829i 0
353.5 0 −0.852797 + 1.50756i 0 0.477579 + 0.477579i 0 −0.988569 + 0.264886i 0 −1.54548 2.57128i 0
353.6 0 −0.411861 + 1.68237i 0 −1.70770 1.70770i 0 3.27107 0.876480i 0 −2.66074 1.38580i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.cn.f 56
3.b odd 2 1 inner 624.2.cn.f 56
4.b odd 2 1 312.2.bp.a 56
12.b even 2 1 312.2.bp.a 56
13.f odd 12 1 inner 624.2.cn.f 56
39.k even 12 1 inner 624.2.cn.f 56
52.l even 12 1 312.2.bp.a 56
156.v odd 12 1 312.2.bp.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bp.a 56 4.b odd 2 1
312.2.bp.a 56 12.b even 2 1
312.2.bp.a 56 52.l even 12 1
312.2.bp.a 56 156.v odd 12 1
624.2.cn.f 56 1.a even 1 1 trivial
624.2.cn.f 56 3.b odd 2 1 inner
624.2.cn.f 56 13.f odd 12 1 inner
624.2.cn.f 56 39.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\(64\!\cdots\!64\)\( T_{5}^{32} + \)\(13\!\cdots\!04\)\( T_{5}^{28} + \)\(17\!\cdots\!33\)\( T_{5}^{24} + \)\(12\!\cdots\!36\)\( T_{5}^{20} + \)\(43\!\cdots\!56\)\( T_{5}^{16} + \)\(50\!\cdots\!76\)\( T_{5}^{12} + \)\(17\!\cdots\!16\)\( T_{5}^{8} + \)\(17\!\cdots\!72\)\( T_{5}^{4} + \)\(89\!\cdots\!56\)\( \)">\(T_{5}^{56} + \cdots\)
\(T_{7}^{28} + \cdots\)