# Properties

 Label 3969.2.a.ba Level $3969$ Weight $2$ Character orbit 3969.a Self dual yes Analytic conductor $31.693$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3969.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.6926245622$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.574857.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 9 x^{2} + 3 x - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{8} + ( -2 - \beta_{3} - \beta_{4} ) q^{10} + ( -1 + \beta_{2} - \beta_{3} ) q^{11} + ( -1 - \beta_{1} + \beta_{4} ) q^{13} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( 3 - \beta_{1} + \beta_{2} ) q^{17} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{19} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{20} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{22} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{23} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{25} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{26} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{31} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{32} + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( 5 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{38} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{40} + ( \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{41} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{43} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{44} + ( -4 + 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{46} + ( 4 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{47} + ( -6 + 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{50} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( 5 - \beta_{1} + 2 \beta_{4} ) q^{53} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{55} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{58} + ( 6 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{59} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{61} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{62} + ( -6 + \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{64} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{65} + ( -1 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{67} + ( 7 - 2 \beta_{1} + 3 \beta_{2} + \beta_{4} ) q^{68} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 4 - \beta_{2} + 3 \beta_{3} ) q^{73} + ( 10 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{74} + ( 3 - 5 \beta_{1} + \beta_{2} - \beta_{4} ) q^{76} + ( 3 - 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{79} + ( 3 + 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{80} + ( 2 - 4 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{82} + ( 1 + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{83} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{86} + ( 4 - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{88} + ( 7 - 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{89} + ( -5 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{92} + ( 3 - 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{94} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{95} + ( -2 - \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 2q^{2} + 4q^{4} + 4q^{5} - 3q^{8} + O(q^{10})$$ $$5q - 2q^{2} + 4q^{4} + 4q^{5} - 3q^{8} - 7q^{10} - 4q^{11} - 8q^{13} - 2q^{16} + 12q^{17} + q^{19} + 5q^{20} + q^{22} - 3q^{23} + q^{25} + 11q^{26} - 7q^{29} - 3q^{31} + 2q^{32} + 3q^{34} + 20q^{38} - 3q^{40} + 5q^{41} + 7q^{43} + 10q^{44} - 3q^{46} + 27q^{47} - 19q^{50} - 10q^{52} + 21q^{53} - 2q^{55} + 10q^{58} + 30q^{59} - 14q^{61} + 6q^{62} - 25q^{64} + 11q^{65} + 2q^{67} + 27q^{68} - 3q^{71} + 15q^{73} + 36q^{74} + 5q^{76} + 4q^{79} + 20q^{80} - 5q^{82} + 9q^{83} + 6q^{85} + 8q^{86} + 18q^{88} + 28q^{89} - 27q^{92} - 3q^{94} + 14q^{95} - 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 9 x^{2} + 3 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu - 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 4 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{2} + 14$$

## 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
 2.38687 1.84124 0.495868 −0.670333 −2.05365
−2.38687 0 3.69714 2.92087 0 0 −4.05086 0 −6.97172
1.2 −1.84124 0 1.39017 −1.33475 0 0 1.12285 0 2.45760
1.3 −0.495868 0 −1.75411 3.69258 0 0 1.86155 0 −1.83103
1.4 0.670333 0 −1.55065 −1.42494 0 0 −2.38012 0 −0.955182
1.5 2.05365 0 2.21746 0.146246 0 0 0.446582 0 0.300337
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 3969.2.a.ba 5
3.b odd 2 1 3969.2.a.bb 5
7.b odd 2 1 3969.2.a.z 5
7.d odd 6 2 567.2.e.f 10
9.c even 3 2 441.2.f.f 10
9.d odd 6 2 1323.2.f.f 10
21.c even 2 1 3969.2.a.bc 5
21.g even 6 2 567.2.e.e 10
63.g even 3 2 441.2.g.f 10
63.h even 3 2 441.2.h.f 10
63.i even 6 2 189.2.h.b 10
63.j odd 6 2 1323.2.h.f 10
63.k odd 6 2 63.2.g.b 10
63.l odd 6 2 441.2.f.e 10
63.n odd 6 2 1323.2.g.f 10
63.o even 6 2 1323.2.f.e 10
63.s even 6 2 189.2.g.b 10
63.t odd 6 2 63.2.h.b yes 10
252.n even 6 2 1008.2.t.i 10
252.r odd 6 2 3024.2.q.i 10
252.bj even 6 2 1008.2.q.i 10
252.bn odd 6 2 3024.2.t.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 63.k odd 6 2
63.2.h.b yes 10 63.t odd 6 2
189.2.g.b 10 63.s even 6 2
189.2.h.b 10 63.i even 6 2
441.2.f.e 10 63.l odd 6 2
441.2.f.f 10 9.c even 3 2
441.2.g.f 10 63.g even 3 2
441.2.h.f 10 63.h even 3 2
567.2.e.e 10 21.g even 6 2
567.2.e.f 10 7.d odd 6 2
1008.2.q.i 10 252.bj even 6 2
1008.2.t.i 10 252.n even 6 2
1323.2.f.e 10 63.o even 6 2
1323.2.f.f 10 9.d odd 6 2
1323.2.g.f 10 63.n odd 6 2
1323.2.h.f 10 63.j odd 6 2
3024.2.q.i 10 252.r odd 6 2
3024.2.t.i 10 252.bn odd 6 2
3969.2.a.z 5 7.b odd 2 1
3969.2.a.ba 5 1.a even 1 1 trivial
3969.2.a.bb 5 3.b odd 2 1
3969.2.a.bc 5 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3969))$$:

 $$T_{2}^{5} + 2 T_{2}^{4} - 5 T_{2}^{3} - 9 T_{2}^{2} + 3 T_{2} + 3$$ $$T_{5}^{5} - 4 T_{5}^{4} - 5 T_{5}^{3} + 18 T_{5}^{2} + 18 T_{5} - 3$$ $$T_{11}^{5} + 4 T_{11}^{4} - 8 T_{11}^{3} - 15 T_{11}^{2} + 12 T_{11} + 15$$ $$T_{13}^{5} + 8 T_{13}^{4} + 13 T_{13}^{3} - 13 T_{13}^{2} - 23 T_{13} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} + 7 T^{3} + 13 T^{4} + 15 T^{5} + 26 T^{6} + 28 T^{7} + 40 T^{8} + 32 T^{9} + 32 T^{10}$$
$3$ 1
$5$ $$1 - 4 T + 20 T^{2} - 62 T^{3} + 193 T^{4} - 423 T^{5} + 965 T^{6} - 1550 T^{7} + 2500 T^{8} - 2500 T^{9} + 3125 T^{10}$$
$7$ 1
$11$ $$1 + 4 T + 47 T^{2} + 161 T^{3} + 958 T^{4} + 2589 T^{5} + 10538 T^{6} + 19481 T^{7} + 62557 T^{8} + 58564 T^{9} + 161051 T^{10}$$
$13$ $$1 + 8 T + 78 T^{2} + 403 T^{3} + 2174 T^{4} + 7779 T^{5} + 28262 T^{6} + 68107 T^{7} + 171366 T^{8} + 228488 T^{9} + 371293 T^{10}$$
$17$ $$1 - 12 T + 130 T^{2} - 876 T^{3} + 5203 T^{4} - 22839 T^{5} + 88451 T^{6} - 253164 T^{7} + 638690 T^{8} - 1002252 T^{9} + 1419857 T^{10}$$
$19$ $$1 - T + 54 T^{2} - 122 T^{3} + 1532 T^{4} - 3483 T^{5} + 29108 T^{6} - 44042 T^{7} + 370386 T^{8} - 130321 T^{9} + 2476099 T^{10}$$
$23$ $$1 + 3 T + 52 T^{2} + 225 T^{3} + 2023 T^{4} + 5565 T^{5} + 46529 T^{6} + 119025 T^{7} + 632684 T^{8} + 839523 T^{9} + 6436343 T^{10}$$
$29$ $$1 + 7 T + 125 T^{2} + 647 T^{3} + 6535 T^{4} + 25761 T^{5} + 189515 T^{6} + 544127 T^{7} + 3048625 T^{8} + 4950967 T^{9} + 20511149 T^{10}$$
$31$ $$1 + 3 T + 134 T^{2} + 308 T^{3} + 7750 T^{4} + 13615 T^{5} + 240250 T^{6} + 295988 T^{7} + 3991994 T^{8} + 2770563 T^{9} + 28629151 T^{10}$$
$37$ $$1 + 89 T^{2} + 280 T^{3} + 3418 T^{4} + 20432 T^{5} + 126466 T^{6} + 383320 T^{7} + 4508117 T^{8} + 69343957 T^{10}$$
$41$ $$1 - 5 T + 161 T^{2} - 769 T^{3} + 11569 T^{4} - 46293 T^{5} + 474329 T^{6} - 1292689 T^{7} + 11096281 T^{8} - 14128805 T^{9} + 115856201 T^{10}$$
$43$ $$1 - 7 T + 126 T^{2} - 474 T^{3} + 5544 T^{4} - 14049 T^{5} + 238392 T^{6} - 876426 T^{7} + 10017882 T^{8} - 23931607 T^{9} + 147008443 T^{10}$$
$47$ $$1 - 27 T + 448 T^{2} - 5169 T^{3} + 48091 T^{4} - 359985 T^{5} + 2260277 T^{6} - 11418321 T^{7} + 46512704 T^{8} - 131751387 T^{9} + 229345007 T^{10}$$
$53$ $$1 - 21 T + 400 T^{2} - 4662 T^{3} + 49132 T^{4} - 375771 T^{5} + 2603996 T^{6} - 13095558 T^{7} + 59550800 T^{8} - 165700101 T^{9} + 418195493 T^{10}$$
$59$ $$1 - 30 T + 601 T^{2} - 8193 T^{3} + 88864 T^{4} - 752289 T^{5} + 5242976 T^{6} - 28519833 T^{7} + 123432779 T^{8} - 363520830 T^{9} + 714924299 T^{10}$$
$61$ $$1 + 14 T + 339 T^{2} + 3409 T^{3} + 43418 T^{4} + 311709 T^{5} + 2648498 T^{6} + 12684889 T^{7} + 76946559 T^{8} + 193841774 T^{9} + 844596301 T^{10}$$
$67$ $$1 - 2 T + 132 T^{2} - 196 T^{3} + 10871 T^{4} - 15429 T^{5} + 728357 T^{6} - 879844 T^{7} + 39700716 T^{8} - 40302242 T^{9} + 1350125107 T^{10}$$
$71$ $$1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10}$$
$73$ $$1 - 15 T + 359 T^{2} - 3943 T^{3} + 53173 T^{4} - 414929 T^{5} + 3881629 T^{6} - 21012247 T^{7} + 139657103 T^{8} - 425973615 T^{9} + 2073071593 T^{10}$$
$79$ $$1 - 4 T + 300 T^{2} - 1488 T^{3} + 39873 T^{4} - 184983 T^{5} + 3149967 T^{6} - 9286608 T^{7} + 147911700 T^{8} - 155800324 T^{9} + 3077056399 T^{10}$$
$83$ $$1 - 9 T + 229 T^{2} - 882 T^{3} + 19849 T^{4} - 37179 T^{5} + 1647467 T^{6} - 6076098 T^{7} + 130939223 T^{8} - 427124889 T^{9} + 3939040643 T^{10}$$
$89$ $$1 - 28 T + 680 T^{2} - 10388 T^{3} + 140263 T^{4} - 1402827 T^{5} + 12483407 T^{6} - 82283348 T^{7} + 479378920 T^{8} - 1756782748 T^{9} + 5584059449 T^{10}$$
$97$ $$1 + 12 T + 341 T^{2} + 2813 T^{3} + 54712 T^{4} + 367945 T^{5} + 5307064 T^{6} + 26467517 T^{7} + 311221493 T^{8} + 1062351372 T^{9} + 8587340257 T^{10}$$