# Properties

 Label 2070.4.a.bj Level $2070$ Weight $4$ Character orbit 2070.a Self dual yes Analytic conductor $122.134$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$122.133953712$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 68 x^{2} - 111 x + 342$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + 5 q^{5} + ( 2 \beta_{1} + \beta_{2} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q + 2 q^{2} + 4 q^{4} + 5 q^{5} + ( 2 \beta_{1} + \beta_{2} ) q^{7} + 8 q^{8} + 10 q^{10} + ( 10 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -5 + 8 \beta_{1} - \beta_{3} ) q^{13} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{14} + 16 q^{16} + ( 6 - 13 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( 14 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{19} + 20 q^{20} + ( 20 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{22} + 23 q^{23} + 25 q^{25} + ( -10 + 16 \beta_{1} - 2 \beta_{3} ) q^{26} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{28} + ( -41 + 14 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{29} + ( 97 + 16 \beta_{1} - 5 \beta_{3} ) q^{31} + 32 q^{32} + ( 12 - 26 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{34} + ( 10 \beta_{1} + 5 \beta_{2} ) q^{35} + ( 116 - 18 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{37} + ( 28 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{38} + 40 q^{40} + ( -121 + \beta_{1} + 10 \beta_{3} ) q^{41} + ( 222 + 24 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{43} + ( 40 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{44} + 46 q^{46} + ( 67 - 44 \beta_{1} + 3 \beta_{2} + 11 \beta_{3} ) q^{47} + ( 411 - 49 \beta_{1} + \beta_{2} + 9 \beta_{3} ) q^{49} + 50 q^{50} + ( -20 + 32 \beta_{1} - 4 \beta_{3} ) q^{52} + ( -146 + 40 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{53} + ( 50 - 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{55} + ( 16 \beta_{1} + 8 \beta_{2} ) q^{56} + ( -82 + 28 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{58} + ( 20 - 10 \beta_{1} - 14 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 290 + 33 \beta_{1} + 7 \beta_{2} - 19 \beta_{3} ) q^{61} + ( 194 + 32 \beta_{1} - 10 \beta_{3} ) q^{62} + 64 q^{64} + ( -25 + 40 \beta_{1} - 5 \beta_{3} ) q^{65} + ( -368 + 28 \beta_{1} - 12 \beta_{3} ) q^{67} + ( 24 - 52 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{68} + ( 20 \beta_{1} + 10 \beta_{2} ) q^{70} + ( -57 - 39 \beta_{1} - 28 \beta_{2} - 4 \beta_{3} ) q^{71} + ( 287 + 44 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{73} + ( 232 - 36 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{74} + ( 56 + 8 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} ) q^{76} + ( 548 + 61 \beta_{1} + 16 \beta_{2} - 17 \beta_{3} ) q^{77} + ( -232 - 72 \beta_{1} - 20 \beta_{2} - 16 \beta_{3} ) q^{79} + 80 q^{80} + ( -242 + 2 \beta_{1} + 20 \beta_{3} ) q^{82} + ( 256 + 26 \beta_{1} - 24 \beta_{2} + 6 \beta_{3} ) q^{83} + ( 30 - 65 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{85} + ( 444 + 48 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{86} + ( 80 - 8 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{88} + ( 450 - 30 \beta_{1} + 16 \beta_{2} + 34 \beta_{3} ) q^{89} + ( 576 - 85 \beta_{1} - 25 \beta_{2} + 51 \beta_{3} ) q^{91} + 92 q^{92} + ( 134 - 88 \beta_{1} + 6 \beta_{2} + 22 \beta_{3} ) q^{94} + ( 70 + 10 \beta_{1} + 15 \beta_{2} - 10 \beta_{3} ) q^{95} + ( -504 - 137 \beta_{1} + 31 \beta_{2} + \beta_{3} ) q^{97} + ( 822 - 98 \beta_{1} + 2 \beta_{2} + 18 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + O(q^{10})$$ $$4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 68 x^{2} - 111 x + 342$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} - 50 \nu + 18$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 2 \nu - 34$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2 \beta_{1} + 34$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3} + 3 \beta_{2} + 56 \beta_{1} + 84$$

## 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
 −6.57209 1.58997 8.73081 −3.74869
2.00000 0 4.00000 5.00000 0 −35.4229 8.00000 0 10.0000
1.2 2.00000 0 4.00000 5.00000 0 −18.5077 8.00000 0 10.0000
1.3 2.00000 0 4.00000 5.00000 0 23.5622 8.00000 0 10.0000
1.4 2.00000 0 4.00000 5.00000 0 29.3684 8.00000 0 10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2070.4.a.bj 4
3.b odd 2 1 230.4.a.h 4
12.b even 2 1 1840.4.a.m 4
15.d odd 2 1 1150.4.a.p 4
15.e even 4 2 1150.4.b.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.h 4 3.b odd 2 1
1150.4.a.p 4 15.d odd 2 1
1150.4.b.n 8 15.e even 4 2
1840.4.a.m 4 12.b even 2 1
2070.4.a.bj 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2070))$$:

 $$T_{7}^{4} + T_{7}^{3} - 1507 T_{7}^{2} + 2618 T_{7} + 453664$$ $$T_{11}^{4} - 39 T_{11}^{3} - 1771 T_{11}^{2} + 94420 T_{11} - 977536$$ $$T_{17}^{4} - 23 T_{17}^{3} - 14205 T_{17}^{2} + 590494 T_{17} + 1048184$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$( -5 + T )^{4}$$
$7$ $$453664 + 2618 T - 1507 T^{2} + T^{3} + T^{4}$$
$11$ $$-977536 + 94420 T - 1771 T^{2} - 39 T^{3} + T^{4}$$
$13$ $$3059002 + 25481 T - 4948 T^{2} + 20 T^{3} + T^{4}$$
$17$ $$1048184 + 590494 T - 14205 T^{2} - 23 T^{3} + T^{4}$$
$19$ $$-78336 + 340080 T - 15029 T^{2} - 53 T^{3} + T^{4}$$
$23$ $$( -23 + T )^{4}$$
$29$ $$1597064 - 1886712 T - 27260 T^{2} + 161 T^{3} + T^{4}$$
$31$ $$-397027152 + 5546709 T + 13076 T^{2} - 388 T^{3} + T^{4}$$
$37$ $$-7921024 - 681856 T + 37920 T^{2} - 466 T^{3} + T^{4}$$
$41$ $$1506099394 - 17572043 T - 36232 T^{2} + 484 T^{3} + T^{4}$$
$43$ $$-5392023552 + 22823424 T + 166752 T^{2} - 894 T^{3} + T^{4}$$
$47$ $$1306137888 + 43523988 T - 230390 T^{2} - 265 T^{3} + T^{4}$$
$53$ $$-3236491568 - 79775808 T - 107352 T^{2} + 576 T^{3} + T^{4}$$
$59$ $$11673423616 + 20596976 T - 243460 T^{2} - 94 T^{3} + T^{4}$$
$61$ $$-42772329400 + 315388730 T - 54057 T^{2} - 1153 T^{3} + T^{4}$$
$67$ $$-10308616192 + 42702336 T + 602032 T^{2} + 1472 T^{3} + T^{4}$$
$71$ $$274201266224 - 93278335 T - 1067322 T^{2} + 200 T^{3} + T^{4}$$
$73$ $$-4754107544 - 9049904 T + 356120 T^{2} - 1147 T^{3} + T^{4}$$
$79$ $$145785325568 - 272612224 T - 923664 T^{2} + 908 T^{3} + T^{4}$$
$83$ $$-178673654272 + 675145440 T - 359444 T^{2} - 1048 T^{3} + T^{4}$$
$89$ $$-969417760000 + 2249185600 T - 492084 T^{2} - 1784 T^{3} + T^{4}$$
$97$ $$-658266694276 - 2746911284 T - 716799 T^{2} + 2047 T^{3} + T^{4}$$