# 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 $8$

# Related objects

## 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(120, [\chi])$$:

 $$T_{7}^{8} - 4T_{7}^{5} + 92T_{7}^{4} - 40T_{7}^{3} + 8T_{7}^{2} + 16T_{7} + 16$$ T7^8 - 4*T7^5 + 92*T7^4 - 40*T7^3 + 8*T7^2 + 16*T7 + 16 $$T_{11}^{8} - 26T_{11}^{6} + 208T_{11}^{4} - 544T_{11}^{2} + 128$$ T11^8 - 26*T11^6 + 208*T11^4 - 544*T11^2 + 128