Properties

Label 120.2.w.c
Level $120$
Weight $2$
Character orbit 120.w
Analytic conductor $0.958$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{6} + 4 q^{10} - 8 q^{12} - 28 q^{15} + 28 q^{16} - 20 q^{18} - 52 q^{22} - 8 q^{25} + 12 q^{28} - 32 q^{30} - 32 q^{31} + 8 q^{33} - 20 q^{36} + 24 q^{40} + 16 q^{42} + 24 q^{46} + 44 q^{48} + 8 q^{52} + 8 q^{55} - 16 q^{57} + 28 q^{58} + 56 q^{60} + 48 q^{63} + 16 q^{66} + 20 q^{70} + 32 q^{72} - 64 q^{73} - 88 q^{76} + 64 q^{78} + 48 q^{81} + 64 q^{82} - 8 q^{87} - 52 q^{88} + 84 q^{90} - 52 q^{96} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.41107 0.0941764i 0.416519 + 1.68122i 1.98226 + 0.265780i −1.62104 + 1.54020i −0.429408 2.41156i −0.361989 + 0.361989i −2.77209 0.561717i −2.65302 + 1.40052i 2.43246 2.02068i
53.2 −1.39193 0.250043i 1.40091 1.01856i 1.87496 + 0.696087i 2.23305 + 0.116202i −2.20465 + 1.06748i −2.29041 + 2.29041i −2.43576 1.43773i 0.925085 2.85381i −3.07920 0.720103i
53.3 −1.30986 + 0.533177i −1.59834 + 0.667305i 1.43144 1.39677i −0.143028 2.23149i 1.73781 1.72627i 0.582772 0.582772i −1.13026 + 2.59278i 2.10941 2.13317i 1.37713 + 2.84667i
53.4 −1.11940 0.864261i −1.72368 0.170116i 0.506107 + 1.93490i 1.36575 + 1.77052i 1.78246 + 1.68013i 2.06963 2.06963i 1.10573 2.60334i 2.94212 + 0.586449i 0.00136610 3.16228i
53.5 −0.864261 1.11940i 1.72368 + 0.170116i −0.506107 + 1.93490i −1.36575 1.77052i −1.29928 2.07651i 2.06963 2.06963i 2.60334 1.10573i 2.94212 + 0.586449i −0.801547 + 3.05901i
53.6 −0.533177 + 1.30986i 0.667305 1.59834i −1.43144 1.39677i −0.143028 2.23149i 1.73781 + 1.72627i 0.582772 0.582772i 2.59278 1.13026i −2.10941 2.13317i 2.99919 + 1.00243i
53.7 −0.250043 1.39193i −1.40091 + 1.01856i −1.87496 + 0.696087i −2.23305 0.116202i 1.76805 + 1.69529i −2.29041 + 2.29041i 1.43773 + 2.43576i 0.925085 2.85381i 0.396613 + 3.13731i
53.8 −0.0941764 1.41107i −0.416519 1.68122i −1.98226 + 0.265780i 1.62104 1.54020i −2.33310 + 0.746071i −0.361989 + 0.361989i 0.561717 + 2.77209i −2.65302 + 1.40052i −2.32601 2.14236i
53.9 0.0941764 + 1.41107i 1.68122 + 0.416519i −1.98226 + 0.265780i −1.62104 + 1.54020i −0.429408 + 2.41156i −0.361989 + 0.361989i −0.561717 2.77209i 2.65302 + 1.40052i −2.32601 2.14236i
53.10 0.250043 + 1.39193i −1.01856 + 1.40091i −1.87496 + 0.696087i 2.23305 + 0.116202i −2.20465 1.06748i −2.29041 + 2.29041i −1.43773 2.43576i −0.925085 2.85381i 0.396613 + 3.13731i
53.11 0.533177 1.30986i 1.59834 0.667305i −1.43144 1.39677i 0.143028 + 2.23149i −0.0218716 2.44939i 0.582772 0.582772i −2.59278 + 1.13026i 2.10941 2.13317i 2.99919 + 1.00243i
53.12 0.864261 + 1.11940i −0.170116 1.72368i −0.506107 + 1.93490i 1.36575 + 1.77052i 1.78246 1.68013i 2.06963 2.06963i −2.60334 + 1.10573i −2.94212 + 0.586449i −0.801547 + 3.05901i
53.13 1.11940 + 0.864261i 0.170116 + 1.72368i 0.506107 + 1.93490i −1.36575 1.77052i −1.29928 + 2.07651i 2.06963 2.06963i −1.10573 + 2.60334i −2.94212 + 0.586449i 0.00136610 3.16228i
53.14 1.30986 0.533177i −0.667305 + 1.59834i 1.43144 1.39677i 0.143028 + 2.23149i −0.0218716 + 2.44939i 0.582772 0.582772i 1.13026 2.59278i −2.10941 2.13317i 1.37713 + 2.84667i
53.15 1.39193 + 0.250043i 1.01856 1.40091i 1.87496 + 0.696087i −2.23305 0.116202i 1.76805 1.69529i −2.29041 + 2.29041i 2.43576 + 1.43773i −0.925085 2.85381i −3.07920 0.720103i
53.16 1.41107 + 0.0941764i −1.68122 0.416519i 1.98226 + 0.265780i 1.62104 1.54020i −2.33310 0.746071i −0.361989 + 0.361989i 2.77209 + 0.561717i 2.65302 + 1.40052i 2.43246 2.02068i
77.1 −1.41107 + 0.0941764i 0.416519 1.68122i 1.98226 0.265780i −1.62104 1.54020i −0.429408 + 2.41156i −0.361989 0.361989i −2.77209 + 0.561717i −2.65302 1.40052i 2.43246 + 2.02068i
77.2 −1.39193 + 0.250043i 1.40091 + 1.01856i 1.87496 0.696087i 2.23305 0.116202i −2.20465 1.06748i −2.29041 2.29041i −2.43576 + 1.43773i 0.925085 + 2.85381i −3.07920 + 0.720103i
77.3 −1.30986 0.533177i −1.59834 0.667305i 1.43144 + 1.39677i −0.143028 + 2.23149i 1.73781 + 1.72627i 0.582772 + 0.582772i −1.13026 2.59278i 2.10941 + 2.13317i 1.37713 2.84667i
77.4 −1.11940 + 0.864261i −1.72368 + 0.170116i 0.506107 1.93490i 1.36575 1.77052i 1.78246 1.68013i 2.06963 + 2.06963i 1.10573 + 2.60334i 2.94212 0.586449i 0.00136610 + 3.16228i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
24.h odd 2 1 inner
40.i odd 4 1 inner
120.w even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.w.c 32
3.b odd 2 1 inner 120.2.w.c 32
4.b odd 2 1 480.2.bi.c 32
5.b even 2 1 600.2.w.j 32
5.c odd 4 1 inner 120.2.w.c 32
5.c odd 4 1 600.2.w.j 32
8.b even 2 1 inner 120.2.w.c 32
8.d odd 2 1 480.2.bi.c 32
12.b even 2 1 480.2.bi.c 32
15.d odd 2 1 600.2.w.j 32
15.e even 4 1 inner 120.2.w.c 32
15.e even 4 1 600.2.w.j 32
20.e even 4 1 480.2.bi.c 32
24.f even 2 1 480.2.bi.c 32
24.h odd 2 1 inner 120.2.w.c 32
40.f even 2 1 600.2.w.j 32
40.i odd 4 1 inner 120.2.w.c 32
40.i odd 4 1 600.2.w.j 32
40.k even 4 1 480.2.bi.c 32
60.l odd 4 1 480.2.bi.c 32
120.i odd 2 1 600.2.w.j 32
120.q odd 4 1 480.2.bi.c 32
120.w even 4 1 inner 120.2.w.c 32
120.w even 4 1 600.2.w.j 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.w.c 32 1.a even 1 1 trivial
120.2.w.c 32 3.b odd 2 1 inner
120.2.w.c 32 5.c odd 4 1 inner
120.2.w.c 32 8.b even 2 1 inner
120.2.w.c 32 15.e even 4 1 inner
120.2.w.c 32 24.h odd 2 1 inner
120.2.w.c 32 40.i odd 4 1 inner
120.2.w.c 32 120.w even 4 1 inner
480.2.bi.c 32 4.b odd 2 1
480.2.bi.c 32 8.d odd 2 1
480.2.bi.c 32 12.b even 2 1
480.2.bi.c 32 20.e even 4 1
480.2.bi.c 32 24.f even 2 1
480.2.bi.c 32 40.k even 4 1
480.2.bi.c 32 60.l odd 4 1
480.2.bi.c 32 120.q odd 4 1
600.2.w.j 32 5.b even 2 1
600.2.w.j 32 5.c odd 4 1
600.2.w.j 32 15.d odd 2 1
600.2.w.j 32 15.e even 4 1
600.2.w.j 32 40.f even 2 1
600.2.w.j 32 40.i odd 4 1
600.2.w.j 32 120.i odd 2 1
600.2.w.j 32 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\):

\( T_{7}^{8} - 4T_{7}^{5} + 92T_{7}^{4} - 40T_{7}^{3} + 8T_{7}^{2} + 16T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 26T_{11}^{6} + 208T_{11}^{4} - 544T_{11}^{2} + 128 \) Copy content Toggle raw display