# Properties

 Label 841.2.b.a Level $841$ Weight $2$ Character orbit 841.b Analytic conductor $6.715$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$841 = 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 841.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.71541880999$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 29) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_1 q^{3} + ( - \beta_{3} - 1) q^{4} + q^{5} + (\beta_{3} + 3) q^{6} - \beta_{3} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - \beta_{3} q^{9}+O(q^{10})$$ q + b1 * q^2 - b1 * q^3 + (-b3 - 1) * q^4 + q^5 + (b3 + 3) * q^6 - b3 * q^7 + (-b2 - 2*b1) * q^8 - b3 * q^9 $$q + \beta_1 q^{2} - \beta_1 q^{3} + ( - \beta_{3} - 1) q^{4} + q^{5} + (\beta_{3} + 3) q^{6} - \beta_{3} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - \beta_{3} q^{9} + \beta_1 q^{10} - \beta_{2} q^{11} + (\beta_{2} + 4 \beta_1) q^{12} + (\beta_{3} + 1) q^{13} + ( - \beta_{2} - 3 \beta_1) q^{14} - \beta_1 q^{15} + 3 q^{16} + 2 \beta_{2} q^{17} + ( - \beta_{2} - 3 \beta_1) q^{18} + ( - 3 \beta_{2} - 3 \beta_1) q^{19} + ( - \beta_{3} - 1) q^{20} + (\beta_{2} + 3 \beta_1) q^{21} - q^{22} + (2 \beta_{3} - 2) q^{23} + ( - 2 \beta_{3} - 5) q^{24} - 4 q^{25} + (\beta_{2} + 4 \beta_1) q^{26} + \beta_{2} q^{27} + (\beta_{3} + 8) q^{28} + (\beta_{3} + 3) q^{30} + (\beta_{2} - 4 \beta_1) q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + q^{33} + 2 q^{34} - \beta_{3} q^{35} + (\beta_{3} + 8) q^{36} + ( - 2 \beta_{2} - 2 \beta_1) q^{37} + (3 \beta_{3} + 6) q^{38} + ( - \beta_{2} - 4 \beta_1) q^{39} + ( - \beta_{2} - 2 \beta_1) q^{40} + (5 \beta_{2} - \beta_1) q^{41} + ( - 3 \beta_{3} - 8) q^{42} + ( - 3 \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{2} - \beta_1) q^{44} - \beta_{3} q^{45} + (2 \beta_{2} + 4 \beta_1) q^{46} + (2 \beta_{2} - \beta_1) q^{47} - 3 \beta_1 q^{48} + q^{49} - 4 \beta_1 q^{50} - 2 q^{51} + ( - 2 \beta_{3} - 9) q^{52} + (3 \beta_{3} + 1) q^{53} + q^{54} - \beta_{2} q^{55} + ( - \beta_{2} + 5 \beta_1) q^{56} + ( - 3 \beta_{3} - 6) q^{57} + ( - 2 \beta_{3} + 2) q^{59} + (\beta_{2} + 4 \beta_1) q^{60} + 2 \beta_1 q^{61} + (4 \beta_{3} + 13) q^{62} + 8 q^{63} + (\beta_{3} + 7) q^{64} + (\beta_{3} + 1) q^{65} + \beta_1 q^{66} - 2 \beta_{3} q^{67} + (4 \beta_{2} + 2 \beta_1) q^{68} + ( - 2 \beta_{2} - 4 \beta_1) q^{69} + ( - \beta_{2} - 3 \beta_1) q^{70} + (\beta_{3} + 6) q^{71} + ( - \beta_{2} + 5 \beta_1) q^{72} + (2 \beta_{2} + 2 \beta_1) q^{73} + (2 \beta_{3} + 4) q^{74} + 4 \beta_1 q^{75} + ( - 3 \beta_{2} + 9 \beta_1) q^{76} + ( - 3 \beta_{2} - \beta_1) q^{77} + (4 \beta_{3} + 11) q^{78} + \beta_1 q^{79} + 3 q^{80} + ( - 3 \beta_{3} - 1) q^{81} + (\beta_{3} + 8) q^{82} + (2 \beta_{3} + 2) q^{83} + ( - \beta_{2} - 11 \beta_1) q^{84} + 2 \beta_{2} q^{85} + (2 \beta_{3} + 3) q^{86} + (\beta_{3} - 1) q^{88} + ( - \beta_{2} + 5 \beta_1) q^{89} + ( - \beta_{2} - 3 \beta_1) q^{90} + ( - \beta_{3} - 8) q^{91} - 14 q^{92} + ( - 4 \beta_{3} - 13) q^{93} + (\beta_{3} + 5) q^{94} + ( - 3 \beta_{2} - 3 \beta_1) q^{95} + ( - \beta_{3} - 1) q^{96} + ( - 5 \beta_{2} + \beta_1) q^{97} + \beta_1 q^{98} + ( - 3 \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 - b1 * q^3 + (-b3 - 1) * q^4 + q^5 + (b3 + 3) * q^6 - b3 * q^7 + (-b2 - 2*b1) * q^8 - b3 * q^9 + b1 * q^10 - b2 * q^11 + (b2 + 4*b1) * q^12 + (b3 + 1) * q^13 + (-b2 - 3*b1) * q^14 - b1 * q^15 + 3 * q^16 + 2*b2 * q^17 + (-b2 - 3*b1) * q^18 + (-3*b2 - 3*b1) * q^19 + (-b3 - 1) * q^20 + (b2 + 3*b1) * q^21 - q^22 + (2*b3 - 2) * q^23 + (-2*b3 - 5) * q^24 - 4 * q^25 + (b2 + 4*b1) * q^26 + b2 * q^27 + (b3 + 8) * q^28 + (b3 + 3) * q^30 + (b2 - 4*b1) * q^31 + (-2*b2 - b1) * q^32 + q^33 + 2 * q^34 - b3 * q^35 + (b3 + 8) * q^36 + (-2*b2 - 2*b1) * q^37 + (3*b3 + 6) * q^38 + (-b2 - 4*b1) * q^39 + (-b2 - 2*b1) * q^40 + (5*b2 - b1) * q^41 + (-3*b3 - 8) * q^42 + (-3*b2 - 2*b1) * q^43 + (-2*b2 - b1) * q^44 - b3 * q^45 + (2*b2 + 4*b1) * q^46 + (2*b2 - b1) * q^47 - 3*b1 * q^48 + q^49 - 4*b1 * q^50 - 2 * q^51 + (-2*b3 - 9) * q^52 + (3*b3 + 1) * q^53 + q^54 - b2 * q^55 + (-b2 + 5*b1) * q^56 + (-3*b3 - 6) * q^57 + (-2*b3 + 2) * q^59 + (b2 + 4*b1) * q^60 + 2*b1 * q^61 + (4*b3 + 13) * q^62 + 8 * q^63 + (b3 + 7) * q^64 + (b3 + 1) * q^65 + b1 * q^66 - 2*b3 * q^67 + (4*b2 + 2*b1) * q^68 + (-2*b2 - 4*b1) * q^69 + (-b2 - 3*b1) * q^70 + (b3 + 6) * q^71 + (-b2 + 5*b1) * q^72 + (2*b2 + 2*b1) * q^73 + (2*b3 + 4) * q^74 + 4*b1 * q^75 + (-3*b2 + 9*b1) * q^76 + (-3*b2 - b1) * q^77 + (4*b3 + 11) * q^78 + b1 * q^79 + 3 * q^80 + (-3*b3 - 1) * q^81 + (b3 + 8) * q^82 + (2*b3 + 2) * q^83 + (-b2 - 11*b1) * q^84 + 2*b2 * q^85 + (2*b3 + 3) * q^86 + (b3 - 1) * q^88 + (-b2 + 5*b1) * q^89 + (-b2 - 3*b1) * q^90 + (-b3 - 8) * q^91 - 14 * q^92 + (-4*b3 - 13) * q^93 + (b3 + 5) * q^94 + (-3*b2 - 3*b1) * q^95 + (-b3 - 1) * q^96 + (-5*b2 + b1) * q^97 + b1 * q^98 + (-3*b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{5} + 12 q^{6}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^5 + 12 * q^6 $$4 q - 4 q^{4} + 4 q^{5} + 12 q^{6} + 4 q^{13} + 12 q^{16} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 20 q^{24} - 16 q^{25} + 32 q^{28} + 12 q^{30} + 4 q^{33} + 8 q^{34} + 32 q^{36} + 24 q^{38} - 32 q^{42} + 4 q^{49} - 8 q^{51} - 36 q^{52} + 4 q^{53} + 4 q^{54} - 24 q^{57} + 8 q^{59} + 52 q^{62} + 32 q^{63} + 28 q^{64} + 4 q^{65} + 24 q^{71} + 16 q^{74} + 44 q^{78} + 12 q^{80} - 4 q^{81} + 32 q^{82} + 8 q^{83} + 12 q^{86} - 4 q^{88} - 32 q^{91} - 56 q^{92} - 52 q^{93} + 20 q^{94} - 4 q^{96}+O(q^{100})$$ 4 * 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 + 32 * q^28 + 12 * 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 - 24 * q^57 + 8 * 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 + 12 * q^86 - 4 * q^88 - 32 * q^91 - 56 * q^92 - 52 * q^93 + 20 * q^94 - 4 * q^96

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
840.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 2.41421i −3.82843 1.00000 5.82843 −2.82843 4.41421i −2.82843 2.41421i
840.2 0.414214i 0.414214i 1.82843 1.00000 0.171573 2.82843 1.58579i 2.82843 0.414214i
840.3 0.414214i 0.414214i 1.82843 1.00000 0.171573 2.82843 1.58579i 2.82843 0.414214i
840.4 2.41421i 2.41421i −3.82843 1.00000 5.82843 −2.82843 4.41421i −2.82843 2.41421i
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.b.a 4
29.b even 2 1 inner 841.2.b.a 4
29.c odd 4 1 29.2.a.a 2
29.c odd 4 1 841.2.a.d 2
29.d even 7 6 841.2.e.k 24
29.e even 14 6 841.2.e.k 24
29.f odd 28 6 841.2.d.f 12
29.f odd 28 6 841.2.d.j 12
87.f even 4 1 261.2.a.d 2
87.f even 4 1 7569.2.a.c 2
116.e even 4 1 464.2.a.h 2
145.e even 4 1 725.2.b.b 4
145.f odd 4 1 725.2.a.b 2
145.j even 4 1 725.2.b.b 4
203.g even 4 1 1421.2.a.j 2
232.k even 4 1 1856.2.a.w 2
232.l odd 4 1 1856.2.a.r 2
319.f even 4 1 3509.2.a.j 2
348.k odd 4 1 4176.2.a.bq 2
377.i odd 4 1 4901.2.a.g 2
435.l even 4 1 6525.2.a.o 2
493.h odd 4 1 8381.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 29.c odd 4 1
261.2.a.d 2 87.f even 4 1
464.2.a.h 2 116.e even 4 1
725.2.a.b 2 145.f odd 4 1
725.2.b.b 4 145.e even 4 1
725.2.b.b 4 145.j even 4 1
841.2.a.d 2 29.c odd 4 1
841.2.b.a 4 1.a even 1 1 trivial
841.2.b.a 4 29.b even 2 1 inner
841.2.d.f 12 29.f odd 28 6
841.2.d.j 12 29.f odd 28 6
841.2.e.k 24 29.d even 7 6
841.2.e.k 24 29.e even 14 6
1421.2.a.j 2 203.g even 4 1
1856.2.a.r 2 232.l odd 4 1
1856.2.a.w 2 232.k even 4 1
3509.2.a.j 2 319.f even 4 1
4176.2.a.bq 2 348.k odd 4 1
4901.2.a.g 2 377.i odd 4 1
6525.2.a.o 2 435.l even 4 1
7569.2.a.c 2 87.f even 4 1
8381.2.a.e 2 493.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(841, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$(T - 1)^{4}$$
$7$ $$(T^{2} - 8)^{2}$$
$11$ $$T^{4} + 6T^{2} + 1$$
$13$ $$(T^{2} - 2 T - 7)^{2}$$
$17$ $$T^{4} + 24T^{2} + 16$$
$19$ $$(T^{2} + 36)^{2}$$
$23$ $$(T^{2} + 4 T - 28)^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 118T^{2} + 1681$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$T^{4} + 176T^{2} + 3136$$
$43$ $$T^{4} + 54T^{2} + 529$$
$47$ $$T^{4} + 38T^{2} + 289$$
$53$ $$(T^{2} - 2 T - 71)^{2}$$
$59$ $$(T^{2} - 4 T - 28)^{2}$$
$61$ $$T^{4} + 24T^{2} + 16$$
$67$ $$(T^{2} - 32)^{2}$$
$71$ $$(T^{2} - 12 T + 28)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$T^{4} + 6T^{2} + 1$$
$83$ $$(T^{2} - 4 T - 28)^{2}$$
$89$ $$T^{4} + 176T^{2} + 3136$$
$97$ $$T^{4} + 176T^{2} + 3136$$