# Properties

 Label 841.2.e.k Level $841$ Weight $2$ Character orbit 841.e Analytic conductor $6.715$ Analytic rank $0$ Dimension $24$ CM no Inner twists $12$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$841 = 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 841.e (of order $$14$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.71541880999$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{14})$$ Twist minimal: no (minimal twist has level 29) Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{4} - 4 q^{5} - 12 q^{6}+O(q^{10})$$ 24 * q + 4 * q^4 - 4 * q^5 - 12 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{4} - 4 q^{5} - 12 q^{6} - 4 q^{13} - 12 q^{16} + 4 q^{20} + 4 q^{22} + 8 q^{23} + 20 q^{24} + 16 q^{25} + 192 q^{28} + 72 q^{30} - 4 q^{33} - 8 q^{34} - 32 q^{36} - 24 q^{38} + 32 q^{42} - 4 q^{49} + 8 q^{51} + 36 q^{52} - 4 q^{53} - 4 q^{54} - 144 q^{57} + 48 q^{59} - 52 q^{62} - 32 q^{63} - 28 q^{64} - 4 q^{65} - 24 q^{71} - 16 q^{74} - 44 q^{78} - 12 q^{80} + 4 q^{81} - 32 q^{82} - 8 q^{83} + 72 q^{86} - 24 q^{88} + 32 q^{91} + 56 q^{92} + 52 q^{93} - 20 q^{94} + 4 q^{96}+O(q^{100})$$ 24 * q + 4 * q^4 - 4 * q^5 - 12 * q^6 - 4 * q^13 - 12 * q^16 + 4 * q^20 + 4 * q^22 + 8 * q^23 + 20 * q^24 + 16 * q^25 + 192 * q^28 + 72 * q^30 - 4 * q^33 - 8 * q^34 - 32 * q^36 - 24 * q^38 + 32 * q^42 - 4 * q^49 + 8 * q^51 + 36 * q^52 - 4 * q^53 - 4 * q^54 - 144 * q^57 + 48 * q^59 - 52 * q^62 - 32 * q^63 - 28 * q^64 - 4 * q^65 - 24 * q^71 - 16 * q^74 - 44 * q^78 - 12 * q^80 + 4 * q^81 - 32 * q^82 - 8 * q^83 + 72 * q^86 - 24 * q^88 + 32 * q^91 + 56 * q^92 + 52 * q^93 - 20 * q^94 + 4 * q^96

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1 −1.88751 + 1.50524i −2.35368 + 0.537213i 0.851905 3.73244i 0.623490 + 0.781831i 3.63396 4.55685i 0.629384 + 2.75751i 1.91526 + 3.97707i 2.54832 1.22721i −2.35368 0.537213i
63.2 −0.323845 + 0.258258i −0.403828 + 0.0921712i −0.406863 + 1.78258i 0.623490 + 0.781831i 0.106974 0.134141i −0.629384 2.75751i −0.688047 1.42874i −2.54832 + 1.22721i −0.403828 0.0921712i
63.3 0.323845 0.258258i 0.403828 0.0921712i −0.406863 + 1.78258i 0.623490 + 0.781831i 0.106974 0.134141i −0.629384 2.75751i 0.688047 + 1.42874i −2.54832 + 1.22721i 0.403828 + 0.0921712i
63.4 1.88751 1.50524i 2.35368 0.537213i 0.851905 3.73244i 0.623490 + 0.781831i 3.63396 4.55685i 0.629384 + 2.75751i −1.91526 3.97707i 2.54832 1.22721i 2.35368 + 0.537213i
196.1 −2.35368 + 0.537213i 1.04749 2.17513i 3.44929 1.66109i −0.222521 0.974928i −1.29695 + 5.68230i 2.54832 + 1.22721i −3.45117 + 2.75222i −1.76350 2.21135i 1.04749 + 2.17513i
196.2 −0.403828 + 0.0921712i 0.179721 0.373194i −1.64736 + 0.793325i −0.222521 0.974928i −0.0381786 + 0.167271i −2.54832 1.22721i 1.23982 0.988722i 1.76350 + 2.21135i 0.179721 + 0.373194i
196.3 0.403828 0.0921712i −0.179721 + 0.373194i −1.64736 + 0.793325i −0.222521 0.974928i −0.0381786 + 0.167271i −2.54832 1.22721i −1.23982 + 0.988722i 1.76350 + 2.21135i −0.179721 0.373194i
196.4 2.35368 0.537213i −1.04749 + 2.17513i 3.44929 1.66109i −0.222521 0.974928i −1.29695 + 5.68230i 2.54832 + 1.22721i 3.45117 2.75222i −1.76350 2.21135i −1.04749 2.17513i
236.1 −2.35368 0.537213i 1.04749 + 2.17513i 3.44929 + 1.66109i −0.222521 + 0.974928i −1.29695 5.68230i 2.54832 1.22721i −3.45117 2.75222i −1.76350 + 2.21135i 1.04749 2.17513i
236.2 −0.403828 0.0921712i 0.179721 + 0.373194i −1.64736 0.793325i −0.222521 + 0.974928i −0.0381786 0.167271i −2.54832 + 1.22721i 1.23982 + 0.988722i 1.76350 2.21135i 0.179721 0.373194i
236.3 0.403828 + 0.0921712i −0.179721 0.373194i −1.64736 0.793325i −0.222521 + 0.974928i −0.0381786 0.167271i −2.54832 + 1.22721i −1.23982 0.988722i 1.76350 2.21135i −0.179721 + 0.373194i
236.4 2.35368 + 0.537213i −1.04749 2.17513i 3.44929 + 1.66109i −0.222521 + 0.974928i −1.29695 5.68230i 2.54832 1.22721i 3.45117 + 2.75222i −1.76350 + 2.21135i −1.04749 + 2.17513i
267.1 −1.88751 1.50524i −2.35368 0.537213i 0.851905 + 3.73244i 0.623490 0.781831i 3.63396 + 4.55685i 0.629384 2.75751i 1.91526 3.97707i 2.54832 + 1.22721i −2.35368 + 0.537213i
267.2 −0.323845 0.258258i −0.403828 0.0921712i −0.406863 1.78258i 0.623490 0.781831i 0.106974 + 0.134141i −0.629384 + 2.75751i −0.688047 + 1.42874i −2.54832 1.22721i −0.403828 + 0.0921712i
267.3 0.323845 + 0.258258i 0.403828 + 0.0921712i −0.406863 1.78258i 0.623490 0.781831i 0.106974 + 0.134141i −0.629384 + 2.75751i 0.688047 1.42874i −2.54832 1.22721i 0.403828 0.0921712i
267.4 1.88751 + 1.50524i 2.35368 + 0.537213i 0.851905 + 3.73244i 0.623490 0.781831i 3.63396 + 4.55685i 0.629384 2.75751i −1.91526 + 3.97707i 2.54832 + 1.22721i 2.35368 0.537213i
270.1 −1.04749 2.17513i 1.88751 1.50524i −2.38699 + 2.99318i −0.900969 + 0.433884i −5.25123 2.52886i −1.76350 2.21135i 4.30354 + 0.982255i 0.629384 2.75751i 1.88751 + 1.50524i
270.2 −0.179721 0.373194i 0.323845 0.258258i 1.14001 1.42952i −0.900969 + 0.433884i −0.154582 0.0744427i 1.76350 + 2.21135i −1.54603 0.352871i −0.629384 + 2.75751i 0.323845 + 0.258258i
270.3 0.179721 + 0.373194i −0.323845 + 0.258258i 1.14001 1.42952i −0.900969 + 0.433884i −0.154582 0.0744427i 1.76350 + 2.21135i 1.54603 + 0.352871i −0.629384 + 2.75751i −0.323845 0.258258i
270.4 1.04749 + 2.17513i −1.88751 + 1.50524i −2.38699 + 2.99318i −0.900969 + 0.433884i −5.25123 2.52886i −1.76350 2.21135i −4.30354 0.982255i 0.629384 2.75751i −1.88751 1.50524i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 651.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner
29.d even 7 5 inner
29.e even 14 5 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.e.k 24
29.b even 2 1 inner 841.2.e.k 24
29.c odd 4 1 841.2.d.f 12
29.c odd 4 1 841.2.d.j 12
29.d even 7 1 841.2.b.a 4
29.d even 7 5 inner 841.2.e.k 24
29.e even 14 1 841.2.b.a 4
29.e even 14 5 inner 841.2.e.k 24
29.f odd 28 1 29.2.a.a 2
29.f odd 28 1 841.2.a.d 2
29.f odd 28 5 841.2.d.f 12
29.f odd 28 5 841.2.d.j 12
87.k even 28 1 261.2.a.d 2
87.k even 28 1 7569.2.a.c 2
116.l even 28 1 464.2.a.h 2
145.o even 28 1 725.2.b.b 4
145.s odd 28 1 725.2.a.b 2
145.t even 28 1 725.2.b.b 4
203.r even 28 1 1421.2.a.j 2
232.u odd 28 1 1856.2.a.r 2
232.v even 28 1 1856.2.a.w 2
319.q even 28 1 3509.2.a.j 2
348.v odd 28 1 4176.2.a.bq 2
377.bb odd 28 1 4901.2.a.g 2
435.bk even 28 1 6525.2.a.o 2
493.y odd 28 1 8381.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 29.f odd 28 1
261.2.a.d 2 87.k even 28 1
464.2.a.h 2 116.l even 28 1
725.2.a.b 2 145.s odd 28 1
725.2.b.b 4 145.o even 28 1
725.2.b.b 4 145.t even 28 1
841.2.a.d 2 29.f odd 28 1
841.2.b.a 4 29.d even 7 1
841.2.b.a 4 29.e even 14 1
841.2.d.f 12 29.c odd 4 1
841.2.d.f 12 29.f odd 28 5
841.2.d.j 12 29.c odd 4 1
841.2.d.j 12 29.f odd 28 5
841.2.e.k 24 1.a even 1 1 trivial
841.2.e.k 24 29.b even 2 1 inner
841.2.e.k 24 29.d even 7 5 inner
841.2.e.k 24 29.e even 14 5 inner
1421.2.a.j 2 203.r even 28 1
1856.2.a.r 2 232.u odd 28 1
1856.2.a.w 2 232.v even 28 1
3509.2.a.j 2 319.q even 28 1
4176.2.a.bq 2 348.v odd 28 1
4901.2.a.g 2 377.bb odd 28 1
6525.2.a.o 2 435.bk even 28 1
7569.2.a.c 2 87.k even 28 1
8381.2.a.e 2 493.y odd 28 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - 6 T_{2}^{22} + 35 T_{2}^{20} - 204 T_{2}^{18} + 1189 T_{2}^{16} - 6930 T_{2}^{14} + 40391 T_{2}^{12} - 6930 T_{2}^{10} + 1189 T_{2}^{8} - 204 T_{2}^{6} + 35 T_{2}^{4} - 6 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(841, [\chi])$$.