# Properties

 Label 980.2.g.a Level $980$ Weight $2$ Character orbit 980.g Analytic conductor $7.825$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$32$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{2} + 4q^{4} - 4q^{8} + 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{2} + 4q^{4} - 4q^{8} + 32q^{9} + 28q^{16} - 8q^{22} - 32q^{25} - 40q^{29} - 4q^{32} + 60q^{36} - 16q^{37} + 36q^{44} - 4q^{46} + 4q^{50} + 16q^{53} + 48q^{57} - 4q^{58} - 28q^{60} + 4q^{64} - 8q^{65} - 8q^{72} - 76q^{74} + 120q^{78} + 72q^{81} - 56q^{86} - 8q^{88} - 4q^{92} + 16q^{93} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
391.1 −1.40582 0.153886i −3.02707 1.95264 + 0.432671i 1.00000i 4.25550 + 0.465823i 0 −2.67847 0.908739i 6.16313 −0.153886 + 1.40582i
391.2 −1.40582 0.153886i 3.02707 1.95264 + 0.432671i 1.00000i −4.25550 0.465823i 0 −2.67847 0.908739i 6.16313 0.153886 1.40582i
391.3 −1.40582 + 0.153886i −3.02707 1.95264 0.432671i 1.00000i 4.25550 0.465823i 0 −2.67847 + 0.908739i 6.16313 −0.153886 1.40582i
391.4 −1.40582 + 0.153886i 3.02707 1.95264 0.432671i 1.00000i −4.25550 + 0.465823i 0 −2.67847 + 0.908739i 6.16313 0.153886 + 1.40582i
391.5 −1.34325 0.442358i −0.901278 1.60864 + 1.18839i 1.00000i 1.21064 + 0.398687i 0 −1.63511 2.30790i −2.18770 0.442358 1.34325i
391.6 −1.34325 0.442358i 0.901278 1.60864 + 1.18839i 1.00000i −1.21064 0.398687i 0 −1.63511 2.30790i −2.18770 −0.442358 + 1.34325i
391.7 −1.34325 + 0.442358i −0.901278 1.60864 1.18839i 1.00000i 1.21064 0.398687i 0 −1.63511 + 2.30790i −2.18770 0.442358 + 1.34325i
391.8 −1.34325 + 0.442358i 0.901278 1.60864 1.18839i 1.00000i −1.21064 + 0.398687i 0 −1.63511 + 2.30790i −2.18770 −0.442358 1.34325i
391.9 −0.976830 1.02265i −1.11294 −0.0916073 + 1.99790i 1.00000i 1.08715 + 1.13814i 0 2.13263 1.85793i −1.76137 1.02265 0.976830i
391.10 −0.976830 1.02265i 1.11294 −0.0916073 + 1.99790i 1.00000i −1.08715 1.13814i 0 2.13263 1.85793i −1.76137 −1.02265 + 0.976830i
391.11 −0.976830 + 1.02265i −1.11294 −0.0916073 1.99790i 1.00000i 1.08715 1.13814i 0 2.13263 + 1.85793i −1.76137 1.02265 + 0.976830i
391.12 −0.976830 + 1.02265i 1.11294 −0.0916073 1.99790i 1.00000i −1.08715 + 1.13814i 0 2.13263 + 1.85793i −1.76137 −1.02265 0.976830i
391.13 −0.431404 1.34681i −2.73718 −1.62778 + 1.16204i 1.00000i 1.18083 + 3.68646i 0 2.26727 + 1.69100i 4.49217 −1.34681 + 0.431404i
391.14 −0.431404 1.34681i 2.73718 −1.62778 + 1.16204i 1.00000i −1.18083 3.68646i 0 2.26727 + 1.69100i 4.49217 1.34681 0.431404i
391.15 −0.431404 + 1.34681i −2.73718 −1.62778 1.16204i 1.00000i 1.18083 3.68646i 0 2.26727 1.69100i 4.49217 −1.34681 0.431404i
391.16 −0.431404 + 1.34681i 2.73718 −1.62778 1.16204i 1.00000i −1.18083 + 3.68646i 0 2.26727 1.69100i 4.49217 1.34681 + 0.431404i
391.17 0.0982654 1.41080i −0.662355 −1.98069 0.277265i 1.00000i −0.0650866 + 0.934447i 0 −0.585797 + 2.76710i −2.56129 1.41080 + 0.0982654i
391.18 0.0982654 1.41080i 0.662355 −1.98069 0.277265i 1.00000i 0.0650866 0.934447i 0 −0.585797 + 2.76710i −2.56129 −1.41080 0.0982654i
391.19 0.0982654 + 1.41080i −0.662355 −1.98069 + 0.277265i 1.00000i −0.0650866 0.934447i 0 −0.585797 2.76710i −2.56129 1.41080 0.0982654i
391.20 0.0982654 + 1.41080i 0.662355 −1.98069 + 0.277265i 1.00000i 0.0650866 + 0.934447i 0 −0.585797 2.76710i −2.56129 −1.41080 + 0.0982654i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.g.a 32
4.b odd 2 1 inner 980.2.g.a 32
7.b odd 2 1 inner 980.2.g.a 32
7.c even 3 1 140.2.o.a 32
7.c even 3 1 980.2.o.f 32
7.d odd 6 1 140.2.o.a 32
7.d odd 6 1 980.2.o.f 32
28.d even 2 1 inner 980.2.g.a 32
28.f even 6 1 140.2.o.a 32
28.f even 6 1 980.2.o.f 32
28.g odd 6 1 140.2.o.a 32
28.g odd 6 1 980.2.o.f 32
35.i odd 6 1 700.2.p.c 32
35.j even 6 1 700.2.p.c 32
35.k even 12 1 700.2.t.c 32
35.k even 12 1 700.2.t.d 32
35.l odd 12 1 700.2.t.c 32
35.l odd 12 1 700.2.t.d 32
140.p odd 6 1 700.2.p.c 32
140.s even 6 1 700.2.p.c 32
140.w even 12 1 700.2.t.c 32
140.w even 12 1 700.2.t.d 32
140.x odd 12 1 700.2.t.c 32
140.x odd 12 1 700.2.t.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.o.a 32 7.c even 3 1
140.2.o.a 32 7.d odd 6 1
140.2.o.a 32 28.f even 6 1
140.2.o.a 32 28.g odd 6 1
700.2.p.c 32 35.i odd 6 1
700.2.p.c 32 35.j even 6 1
700.2.p.c 32 140.p odd 6 1
700.2.p.c 32 140.s even 6 1
700.2.t.c 32 35.k even 12 1
700.2.t.c 32 35.l odd 12 1
700.2.t.c 32 140.w even 12 1
700.2.t.c 32 140.x odd 12 1
700.2.t.d 32 35.k even 12 1
700.2.t.d 32 35.l odd 12 1
700.2.t.d 32 140.w even 12 1
700.2.t.d 32 140.x odd 12 1
980.2.g.a 32 1.a even 1 1 trivial
980.2.g.a 32 4.b odd 2 1 inner
980.2.g.a 32 7.b odd 2 1 inner
980.2.g.a 32 28.d even 2 1 inner
980.2.o.f 32 7.c even 3 1
980.2.o.f 32 7.d odd 6 1
980.2.o.f 32 28.f even 6 1
980.2.o.f 32 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(980, [\chi])$$.