# Properties

 Label 3380.2.a.s Level $3380$ Weight $2$ Character orbit 3380.a Self dual yes Analytic conductor $26.989$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9894358832$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{9} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{11} -\beta_{1} q^{15} + ( 1 + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{17} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{19} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{23} + q^{25} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 3 \beta_{7} + \beta_{8} ) q^{27} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{29} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{31} + ( 3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{33} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{35} + ( \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{37} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{41} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{43} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{45} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} ) q^{47} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{7} ) q^{49} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} ) q^{51} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{53} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{55} + ( 4 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{5} + 3 \beta_{7} + \beta_{8} ) q^{57} + ( -2 \beta_{1} - \beta_{2} - 4 \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{59} + ( 3 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{61} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{63} + ( 4 + 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + 6 \beta_{7} + 2 \beta_{8} ) q^{67} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + 6 \beta_{7} + \beta_{8} ) q^{69} + ( -1 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{8} ) q^{71} + ( -4 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{73} -\beta_{1} q^{75} + ( 7 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{77} + ( 2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{79} + ( 8 - \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{81} + ( 6 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{83} + ( 1 + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{85} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 8 \beta_{4} + \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{87} + ( -5 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{8} ) q^{89} + ( -1 + 3 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + \beta_{6} + 6 \beta_{7} + 2 \beta_{8} ) q^{93} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{95} + ( -3 - 4 \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - 6 \beta_{7} - 2 \beta_{8} ) q^{97} + ( -14 - 2 \beta_{1} + 5 \beta_{2} - 8 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9} + O(q^{10})$$ $$9 q - q^{3} + 9 q^{5} + q^{7} + 12 q^{9} - 7 q^{11} - q^{15} + 13 q^{17} - 4 q^{19} - 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} + 13 q^{31} + 34 q^{33} + q^{35} + q^{37} - 6 q^{41} + q^{43} + 12 q^{45} - 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} + 38 q^{57} + 15 q^{59} + 21 q^{61} - 17 q^{63} - 7 q^{67} + 15 q^{69} - 7 q^{71} - 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} + 45 q^{83} + 13 q^{85} + 28 q^{87} - 41 q^{89} - 11 q^{93} - 4 q^{95} + 8 q^{97} - 81 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{8} + 42 \nu^{7} - 241 \nu^{6} - 394 \nu^{5} + 1964 \nu^{4} + 214 \nu^{3} - 3215 \nu^{2} + 1620 \nu + 226$$$$)/338$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{8} + 93 \nu^{7} - 256 \nu^{6} - 969 \nu^{5} + 2683 \nu^{4} + 1512 \nu^{3} - 5103 \nu^{2} + 1873 \nu - 79$$$$)/338$$ $$\beta_{4}$$ $$=$$ $$($$$$-33 \nu^{8} + 27 \nu^{7} + 678 \nu^{6} - 543 \nu^{5} - 4073 \nu^{4} + 3590 \nu^{3} + 6347 \nu^{2} - 6805 \nu + 773$$$$)/338$$ $$\beta_{5}$$ $$=$$ $$($$$$84 \nu^{8} - 38 \nu^{7} - 1649 \nu^{6} + 614 \nu^{5} + 9369 \nu^{4} - 3976 \nu^{3} - 13452 \nu^{2} + 8288 \nu - 1645$$$$)/338$$ $$\beta_{6}$$ $$=$$ $$($$$$108 \nu^{8} - 73 \nu^{7} - 2096 \nu^{6} + 1055 \nu^{5} + 12070 \nu^{4} - 5450 \nu^{3} - 19082 \nu^{2} + 9473 \nu + 82$$$$)/338$$ $$\beta_{7}$$ $$=$$ $$($$$$112 \nu^{8} - 107 \nu^{7} - 2086 \nu^{6} + 1551 \nu^{5} + 11478 \nu^{4} - 7780 \nu^{3} - 16922 \nu^{2} + 13473 \nu - 1292$$$$)/338$$ $$\beta_{8}$$ $$=$$ $$($$$$-323 \nu^{8} + 295 \nu^{7} + 6206 \nu^{6} - 4731 \nu^{5} - 35175 \nu^{4} + 25992 \nu^{3} + 53223 \nu^{2} - 44995 \nu + 3551$$$$)/338$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{8} - 3 \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{2} + 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{8} - 10 \beta_{7} + 8 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 14 \beta_{2} - \beta_{1} + 35$$ $$\nu^{5}$$ $$=$$ $$-14 \beta_{8} - 38 \beta_{7} - 3 \beta_{6} + 13 \beta_{5} + 28 \beta_{4} + 5 \beta_{3} - 21 \beta_{2} + 53 \beta_{1} + 17$$ $$\nu^{6}$$ $$=$$ $$-21 \beta_{8} - 106 \beta_{7} - 8 \beta_{6} + 74 \beta_{5} - 18 \beta_{4} + 47 \beta_{3} - 162 \beta_{2} - 7 \beta_{1} + 338$$ $$\nu^{7}$$ $$=$$ $$-162 \beta_{8} - 418 \beta_{7} - 55 \beta_{6} + 155 \beta_{5} + 330 \beta_{4} + 105 \beta_{3} - 319 \beta_{2} + 428 \beta_{1} + 355$$ $$\nu^{8}$$ $$=$$ $$-319 \beta_{8} - 1179 \beta_{7} - 160 \beta_{6} + 747 \beta_{5} + 195 \beta_{4} + 599 \beta_{3} - 1817 \beta_{2} + 13 \beta_{1} + 3428$$

## 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
 3.40622 2.52382 1.40883 0.715841 0.161999 0.0545075 −1.65879 −2.79462 −2.81781
0 −3.40622 0 1.00000 0 2.33790 0 8.60234 0
1.2 0 −2.52382 0 1.00000 0 −4.54727 0 3.36968 0
1.3 0 −1.40883 0 1.00000 0 −2.17051 0 −1.01520 0
1.4 0 −0.715841 0 1.00000 0 3.76323 0 −2.48757 0
1.5 0 −0.161999 0 1.00000 0 1.89771 0 −2.97376 0
1.6 0 −0.0545075 0 1.00000 0 3.71818 0 −2.99703 0
1.7 0 1.65879 0 1.00000 0 −4.25414 0 −0.248414 0
1.8 0 2.79462 0 1.00000 0 1.22908 0 4.80991 0
1.9 0 2.81781 0 1.00000 0 −0.974186 0 4.94004 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 3380.2.a.s yes 9
13.b even 2 1 3380.2.a.r 9
13.d odd 4 2 3380.2.f.j 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3380.2.a.r 9 13.b even 2 1
3380.2.a.s yes 9 1.a even 1 1 trivial
3380.2.f.j 18 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3380))$$:

 $$T_{3}^{9} + \cdots$$ $$T_{7}^{9} - \cdots$$ $$T_{19}^{9} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9}$$
$3$ $$-1 - 26 T - 149 T^{2} - 153 T^{3} + 87 T^{4} + 106 T^{5} - 16 T^{6} - 19 T^{7} + T^{8} + T^{9}$$
$5$ $$( -1 + T )^{9}$$
$7$ $$-3121 + 1126 T + 3909 T^{2} - 1877 T^{3} - 951 T^{4} + 518 T^{5} + 60 T^{6} - 41 T^{7} - T^{8} + T^{9}$$
$11$ $$-94016 - 146176 T - 59760 T^{2} + 9440 T^{3} + 9992 T^{4} + 692 T^{5} - 473 T^{6} - 58 T^{7} + 7 T^{8} + T^{9}$$
$13$ $$T^{9}$$
$17$ $$86528 - 332800 T + 85120 T^{2} + 52000 T^{3} - 17496 T^{4} - 1492 T^{5} + 877 T^{6} - 26 T^{7} - 13 T^{8} + T^{9}$$
$19$ $$-64 - 320 T + 3600 T^{2} - 7568 T^{3} + 3412 T^{4} + 2116 T^{5} - 253 T^{6} - 87 T^{7} + 4 T^{8} + T^{9}$$
$23$ $$46171 - 29147 T - 35592 T^{2} + 32719 T^{3} - 4363 T^{4} - 2763 T^{5} + 873 T^{6} - 34 T^{7} - 12 T^{8} + T^{9}$$
$29$ $$-96559 + 84569 T + 253348 T^{2} + 53173 T^{3} - 32345 T^{4} - 3127 T^{5} + 1579 T^{6} - 42 T^{7} - 16 T^{8} + T^{9}$$
$31$ $$-105664 + 982272 T + 720592 T^{2} - 30960 T^{3} - 67008 T^{4} + 3144 T^{5} + 1725 T^{6} - 112 T^{7} - 13 T^{8} + T^{9}$$
$37$ $$18752 + 53568 T - 15792 T^{2} - 97136 T^{3} - 12968 T^{4} + 7832 T^{5} + 321 T^{6} - 169 T^{7} - T^{8} + T^{9}$$
$41$ $$23863181 + 8113743 T - 2724582 T^{2} - 825051 T^{3} + 96715 T^{4} + 23743 T^{5} - 1311 T^{6} - 264 T^{7} + 6 T^{8} + T^{9}$$
$43$ $$-9484117 + 5928484 T + 1527395 T^{2} - 681337 T^{3} - 41569 T^{4} + 20630 T^{5} + 380 T^{6} - 243 T^{7} - T^{8} + T^{9}$$
$47$ $$25493819 + 5197257 T - 3326166 T^{2} - 780037 T^{3} + 83501 T^{4} + 23837 T^{5} - 725 T^{6} - 266 T^{7} + 2 T^{8} + T^{9}$$
$53$ $$118208 + 231296 T - 145232 T^{2} - 70832 T^{3} + 57808 T^{4} - 10400 T^{5} - 325 T^{6} + 287 T^{7} - 30 T^{8} + T^{9}$$
$59$ $$-512 - 5376 T + 11072 T^{2} + 21664 T^{3} - 7440 T^{4} - 10296 T^{5} + 2775 T^{6} - 121 T^{7} - 15 T^{8} + T^{9}$$
$61$ $$1253057 + 3967046 T + 3193811 T^{2} + 116995 T^{3} - 187917 T^{4} - 4048 T^{5} + 3888 T^{6} - 99 T^{7} - 21 T^{8} + T^{9}$$
$67$ $$4851601 - 10949069 T - 5222685 T^{2} + 164623 T^{3} + 304407 T^{4} + 25721 T^{5} - 2890 T^{6} - 332 T^{7} + 7 T^{8} + T^{9}$$
$71$ $$110282432 + 14212672 T - 8691152 T^{2} - 1127472 T^{3} + 228392 T^{4} + 29840 T^{5} - 2321 T^{6} - 310 T^{7} + 7 T^{8} + T^{9}$$
$73$ $$-7046656 - 49346816 T - 12267904 T^{2} + 2750976 T^{3} + 550048 T^{4} - 26112 T^{5} - 7080 T^{6} - 64 T^{7} + 28 T^{8} + T^{9}$$
$79$ $$342428864 - 98522688 T - 10412416 T^{2} + 5714384 T^{3} - 334396 T^{4} - 61800 T^{5} + 7135 T^{6} + 46 T^{7} - 31 T^{8} + T^{9}$$
$83$ $$-65550407 + 56345154 T - 14665643 T^{2} + 35267 T^{3} + 560869 T^{4} - 79106 T^{5} + 720 T^{6} + 601 T^{7} - 45 T^{8} + T^{9}$$
$89$ $$54990949 + 69505827 T + 26276881 T^{2} + 2680497 T^{3} - 357987 T^{4} - 86531 T^{5} - 3100 T^{6} + 438 T^{7} + 41 T^{8} + T^{9}$$
$97$ $$7849472 + 22536192 T - 4079104 T^{2} - 2857376 T^{3} - 62344 T^{4} + 56012 T^{5} + 1859 T^{6} - 401 T^{7} - 8 T^{8} + T^{9}$$