# Properties

 Label 441.4.a.u Level $441$ Weight $4$ Character orbit 441.a Self dual yes Analytic conductor $26.020$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.0198423125$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{65})$$ Defining polynomial: $$x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{2} + ( 9 + \beta_{1} ) q^{4} + \beta_{3} q^{5} + ( -17 - \beta_{1} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{2} + ( 9 + \beta_{1} ) q^{4} + \beta_{3} q^{5} + ( -17 - \beta_{1} ) q^{8} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{10} + ( -22 + 6 \beta_{1} ) q^{11} + ( 6 \beta_{2} + \beta_{3} ) q^{13} + ( -39 + 9 \beta_{1} ) q^{16} + ( \beta_{2} - 9 \beta_{3} ) q^{17} + ( -11 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 2 \beta_{2} + 12 \beta_{3} ) q^{20} + ( -74 + 22 \beta_{1} ) q^{22} + ( -92 - 8 \beta_{1} ) q^{23} + ( -21 + 22 \beta_{1} ) q^{25} + ( 16 \beta_{2} - 16 \beta_{3} ) q^{26} + ( -58 + 14 \beta_{1} ) q^{29} + ( -38 \beta_{2} + 4 \beta_{3} ) q^{31} + ( 31 + 47 \beta_{1} ) q^{32} + ( 21 \beta_{2} + 34 \beta_{3} ) q^{34} + ( 50 - 6 \beta_{1} ) q^{37} + ( -25 \beta_{2} + 38 \beta_{3} ) q^{38} + ( -2 \beta_{2} - 20 \beta_{3} ) q^{40} + ( 31 \beta_{2} - 3 \beta_{3} ) q^{41} + ( 170 + 70 \beta_{1} ) q^{43} + ( -102 + 26 \beta_{1} ) q^{44} + ( 220 + 92 \beta_{1} ) q^{46} + ( 26 \beta_{2} - 32 \beta_{3} ) q^{47} + ( -331 + 21 \beta_{1} ) q^{50} + ( 32 \beta_{2} + 24 \beta_{3} ) q^{52} + ( -22 - 36 \beta_{1} ) q^{53} + ( 12 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -166 + 58 \beta_{1} ) q^{58} + ( -49 \beta_{2} + 20 \beta_{3} ) q^{59} + ( 68 \beta_{2} - 27 \beta_{3} ) q^{61} + ( -122 \beta_{2} + 60 \beta_{3} ) q^{62} + ( -471 - 103 \beta_{1} ) q^{64} + ( 224 + 70 \beta_{1} ) q^{65} + ( -524 - 76 \beta_{1} ) q^{67} + ( -13 \beta_{2} - 106 \beta_{3} ) q^{68} + ( -548 + 28 \beta_{1} ) q^{71} + ( -37 \beta_{2} - 25 \beta_{3} ) q^{73} + ( 46 - 50 \beta_{1} ) q^{74} + ( -63 \beta_{2} - 70 \beta_{3} ) q^{76} + ( -356 - 188 \beta_{1} ) q^{79} + ( 18 \beta_{2} - 12 \beta_{3} ) q^{80} + ( 99 \beta_{2} - 50 \beta_{3} ) q^{82} + ( -\beta_{2} + 8 \beta_{3} ) q^{83} + ( -916 - 190 \beta_{1} ) q^{85} + ( -1290 - 170 \beta_{1} ) q^{86} + ( 278 - 74 \beta_{1} ) q^{88} + ( 157 \beta_{2} + 75 \beta_{3} ) q^{89} + ( -956 - 156 \beta_{1} ) q^{92} + ( 142 \beta_{2} + 76 \beta_{3} ) q^{94} + ( -636 - 176 \beta_{1} ) q^{95} + ( -189 \beta_{2} + 91 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 34q^{4} - 66q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 34q^{4} - 66q^{8} - 100q^{11} - 174q^{16} - 340q^{22} - 352q^{23} - 128q^{25} - 260q^{29} + 30q^{32} + 212q^{37} + 540q^{43} - 460q^{44} + 696q^{46} - 1366q^{50} - 16q^{53} - 780q^{58} - 1678q^{64} + 756q^{65} - 1944q^{67} - 2248q^{71} + 284q^{74} - 1048q^{79} - 3284q^{85} - 4820q^{86} + 1260q^{88} - 3512q^{92} - 2192q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + 3 \nu^{2} + 100 \nu - 79$$$$)/57$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 11 \nu^{2} + 12 \nu - 182$$$$)/19$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{3} + 12 \nu^{2} - 265 \nu - 392$$$$)/57$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 7 \beta_{1} + 7$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} + 10 \beta_{2} + 7 \beta_{1} + 133$$$$)/7$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{3} - 5 \beta_{2} + 23 \beta_{1} + 39$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.11692 5.94534 −4.94534 −2.11692
−4.53113 0 12.5311 −13.4791 0 0 −20.5311 0 61.0753
1.2 −4.53113 0 12.5311 13.4791 0 0 −20.5311 0 −61.0753
1.3 3.53113 0 4.46887 −2.07730 0 0 −12.4689 0 −7.33521
1.4 3.53113 0 4.46887 2.07730 0 0 −12.4689 0 7.33521
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.a.u 4
3.b odd 2 1 49.4.a.e 4
7.b odd 2 1 inner 441.4.a.u 4
7.c even 3 2 441.4.e.y 8
7.d odd 6 2 441.4.e.y 8
12.b even 2 1 784.4.a.bf 4
15.d odd 2 1 1225.4.a.bb 4
21.c even 2 1 49.4.a.e 4
21.g even 6 2 49.4.c.e 8
21.h odd 6 2 49.4.c.e 8
84.h odd 2 1 784.4.a.bf 4
105.g even 2 1 1225.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 3.b odd 2 1
49.4.a.e 4 21.c even 2 1
49.4.c.e 8 21.g even 6 2
49.4.c.e 8 21.h odd 6 2
441.4.a.u 4 1.a even 1 1 trivial
441.4.a.u 4 7.b odd 2 1 inner
441.4.e.y 8 7.c even 3 2
441.4.e.y 8 7.d odd 6 2
784.4.a.bf 4 12.b even 2 1
784.4.a.bf 4 84.h odd 2 1
1225.4.a.bb 4 15.d odd 2 1
1225.4.a.bb 4 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} + T_{2} - 16$$ $$T_{5}^{4} - 186 T_{5}^{2} + 784$$ $$T_{13}^{4} - 3234 T_{13}^{2} + 2458624$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -16 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$784 - 186 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 40 + 50 T + T^{2} )^{2}$$
$13$ $$2458624 - 3234 T^{2} + T^{4}$$
$17$ $$9746884 - 14564 T^{2} + T^{4}$$
$19$ $$52591504 - 14746 T^{2} + T^{4}$$
$23$ $$( 6704 + 176 T + T^{2} )^{2}$$
$29$ $$( 1040 + 130 T + T^{2} )^{2}$$
$31$ $$629407744 - 100104 T^{2} + T^{4}$$
$37$ $$( 2224 - 106 T + T^{2} )^{2}$$
$41$ $$307721764 - 66836 T^{2} + T^{4}$$
$43$ $$( -61400 - 270 T + T^{2} )^{2}$$
$47$ $$8332038400 - 187240 T^{2} + T^{4}$$
$53$ $$( -21044 + 8 T + T^{2} )^{2}$$
$59$ $$1600960144 - 189354 T^{2} + T^{4}$$
$61$ $$5022273424 - 360266 T^{2} + T^{4}$$
$67$ $$( 142336 + 972 T + T^{2} )^{2}$$
$71$ $$( 303104 + 1124 T + T^{2} )^{2}$$
$73$ $$12428236324 - 276756 T^{2} + T^{4}$$
$79$ $$( -505696 + 524 T + T^{2} )^{2}$$
$83$ $$6492304 - 11466 T^{2} + T^{4}$$
$89$ $$2844693770884 - 3623876 T^{2} + T^{4}$$
$97$ $$841222821124 - 3082884 T^{2} + T^{4}$$