# Properties

 Label 161.2.a.d Level 161 Weight 2 Character orbit 161.a Self dual yes Analytic conductor 1.286 Analytic rank 0 Dimension 5 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$161 = 7 \cdot 23$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 161.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.28559147254$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.2147108.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -\beta_{2} + \beta_{4} ) q^{6} + q^{7} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 3 - \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( -\beta_{2} + \beta_{4} ) q^{6} + q^{7} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 3 - \beta_{1} - \beta_{2} ) q^{9} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{12} + ( -1 + \beta_{3} + \beta_{4} ) q^{13} + \beta_{1} q^{14} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{15} + ( 1 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -3 + 2 \beta_{2} + \beta_{4} ) q^{17} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{18} + ( 2 - 2 \beta_{1} ) q^{19} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} ) q^{20} + \beta_{3} q^{21} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{22} - q^{23} + ( -8 + 2 \beta_{1} - \beta_{3} ) q^{24} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{25} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{26} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{27} + ( 2 + \beta_{2} ) q^{28} + ( \beta_{1} - 3 \beta_{2} ) q^{29} + ( -11 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{30} + ( 6 - \beta_{3} ) q^{31} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{32} + ( -4 + 2 \beta_{4} ) q^{33} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{34} + ( -1 - \beta_{4} ) q^{35} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{36} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{38} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} + ( -9 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{4} ) q^{40} + ( 1 + \beta_{3} - \beta_{4} ) q^{41} + ( -\beta_{2} + \beta_{4} ) q^{42} + ( -2 + 2 \beta_{4} ) q^{43} + ( -5 + 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{44} + ( -5 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{45} -\beta_{1} q^{46} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{47} + ( 6 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{48} + q^{49} + ( -3 + 4 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{50} + ( -1 - \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{4} ) q^{51} + ( -4 + \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{52} + ( 4 - 2 \beta_{2} ) q^{53} + ( 7 - \beta_{2} - 2 \beta_{3} ) q^{54} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{55} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{56} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{57} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{58} + ( 5 - 2 \beta_{1} - \beta_{4} ) q^{59} + ( 7 - 5 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{60} + ( -3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{61} + ( 6 \beta_{1} + \beta_{2} - \beta_{4} ) q^{62} + ( 3 - \beta_{1} - \beta_{2} ) q^{63} + ( 5 + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{64} + ( -4 - 2 \beta_{1} + 2 \beta_{4} ) q^{65} + ( 2 - 6 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{66} + ( -3 + 5 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{67} + ( 1 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{68} -\beta_{3} q^{69} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{70} + ( 1 + 3 \beta_{3} + \beta_{4} ) q^{71} + ( -6 - 2 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - \beta_{4} ) q^{72} + ( 1 - 4 \beta_{2} - \beta_{3} - \beta_{4} ) q^{73} + ( 8 + 2 \beta_{4} ) q^{74} + ( -8 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{75} + ( 4 - 6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{76} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{77} + ( 7 + 2 \beta_{1} - \beta_{2} - 4 \beta_{4} ) q^{78} + ( 7 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{79} + ( 3 - 8 \beta_{1} - 6 \beta_{2} - 3 \beta_{4} ) q^{80} + ( -\beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{81} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 2 + 2 \beta_{4} ) q^{83} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{84} + ( 1 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{86} + ( -3 + 6 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{87} + ( 3 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{88} + ( -5 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{89} + ( 7 + 2 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} ) q^{90} + ( -1 + \beta_{3} + \beta_{4} ) q^{91} + ( -2 - \beta_{2} ) q^{92} + ( -6 + \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{93} + ( -8 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{94} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( -1 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{96} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} ) q^{97} + \beta_{1} q^{98} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 2q^{2} + 12q^{4} - 4q^{5} - 3q^{6} + 5q^{7} + 3q^{8} + 11q^{9} + O(q^{10})$$ $$5q + 2q^{2} + 12q^{4} - 4q^{5} - 3q^{6} + 5q^{7} + 3q^{8} + 11q^{9} - 8q^{10} - 4q^{11} + 3q^{12} - 6q^{13} + 2q^{14} + 10q^{15} + 10q^{16} - 12q^{17} - 19q^{18} + 6q^{19} + 14q^{22} - 5q^{23} - 36q^{24} + 19q^{25} + q^{26} + 12q^{28} - 4q^{29} - 48q^{30} + 30q^{31} + 8q^{32} - 22q^{33} + 6q^{34} - 4q^{35} - q^{36} + 4q^{37} - 40q^{38} + 16q^{39} - 50q^{40} + 6q^{41} - 3q^{42} - 12q^{43} - 26q^{44} - 12q^{45} - 2q^{46} + 10q^{47} + 25q^{48} + 5q^{49} - 2q^{50} - 4q^{51} - 21q^{52} + 16q^{53} + 33q^{54} + 18q^{55} + 3q^{56} + 6q^{57} + 13q^{58} + 22q^{59} + 30q^{60} - 18q^{61} + 15q^{62} + 11q^{63} + 25q^{64} - 26q^{65} + 4q^{66} - 2q^{67} + 12q^{68} - 8q^{70} + 4q^{71} - 41q^{72} - 2q^{73} + 38q^{74} - 30q^{75} + 10q^{76} - 4q^{77} + 41q^{78} + 30q^{79} - 10q^{80} - 3q^{81} - 7q^{82} + 8q^{83} + 3q^{84} - 12q^{85} + 8q^{86} - 12q^{87} + 4q^{88} - 20q^{89} + 34q^{90} - 6q^{91} - 12q^{92} - 26q^{93} - 25q^{94} + 8q^{95} - q^{96} - 12q^{97} + 2q^{98} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 9 x^{3} + 17 x^{2} + 16 x - 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} - 8 \nu^{2} + 5 \nu + 11$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + \nu^{3} - 10 \nu^{2} - 5 \nu + 19$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 9 \beta_{2} + 21$$

## 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
 −2.54577 −1.50216 1.23828 2.11948 2.69017
−2.54577 2.46268 4.48096 2.78847 −6.26943 1.00000 −6.31597 3.06481 −7.09882
1.2 −1.50216 −3.04067 0.256481 −3.82405 4.56757 1.00000 2.61904 6.24568 5.74433
1.3 1.23828 2.68857 −0.466664 −1.86253 3.32920 1.00000 −3.05442 4.22838 −2.30633
1.4 2.11948 −1.84074 2.49221 2.40920 −3.90141 1.00000 1.04322 0.388311 5.10626
1.5 2.69017 −0.269842 5.23702 −3.51109 −0.725921 1.00000 8.70812 −2.92719 −9.44544
 $$n$$: 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

## 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 161.2.a.d 5
3.b odd 2 1 1449.2.a.r 5
4.b odd 2 1 2576.2.a.bd 5
5.b even 2 1 4025.2.a.p 5
7.b odd 2 1 1127.2.a.h 5
23.b odd 2 1 3703.2.a.j 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.d 5 1.a even 1 1 trivial
1127.2.a.h 5 7.b odd 2 1
1449.2.a.r 5 3.b odd 2 1
2576.2.a.bd 5 4.b odd 2 1
3703.2.a.j 5 23.b odd 2 1
4025.2.a.p 5 5.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$23$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 2 T_{2}^{4} - 9 T_{2}^{3} + 17 T_{2}^{2} + 16 T_{2} - 27$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(161))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + T^{2} + T^{3} + 2 T^{4} - 7 T^{5} + 4 T^{6} + 4 T^{7} + 8 T^{8} - 32 T^{9} + 32 T^{10}$$
$3$ $$1 + 2 T^{2} + 11 T^{4} + 10 T^{5} + 33 T^{6} + 54 T^{8} + 243 T^{10}$$
$5$ $$1 + 4 T + 11 T^{2} + 26 T^{3} + 92 T^{4} + 228 T^{5} + 460 T^{6} + 650 T^{7} + 1375 T^{8} + 2500 T^{9} + 3125 T^{10}$$
$7$ $$( 1 - T )^{5}$$
$11$ $$1 + 4 T + 27 T^{2} + 28 T^{3} + 126 T^{4} - 400 T^{5} + 1386 T^{6} + 3388 T^{7} + 35937 T^{8} + 58564 T^{9} + 161051 T^{10}$$
$13$ $$1 + 6 T + 56 T^{2} + 266 T^{3} + 1351 T^{4} + 4944 T^{5} + 17563 T^{6} + 44954 T^{7} + 123032 T^{8} + 171366 T^{9} + 371293 T^{10}$$
$17$ $$1 + 12 T + 91 T^{2} + 430 T^{3} + 1692 T^{4} + 6148 T^{5} + 28764 T^{6} + 124270 T^{7} + 447083 T^{8} + 1002252 T^{9} + 1419857 T^{10}$$
$19$ $$1 - 6 T + 67 T^{2} - 360 T^{3} + 2334 T^{4} - 9220 T^{5} + 44346 T^{6} - 129960 T^{7} + 459553 T^{8} - 781926 T^{9} + 2476099 T^{10}$$
$23$ $$( 1 + T )^{5}$$
$29$ $$1 + 4 T + 34 T^{2} + 214 T^{3} + 1217 T^{4} + 4232 T^{5} + 35293 T^{6} + 179974 T^{7} + 829226 T^{8} + 2829124 T^{9} + 20511149 T^{10}$$
$31$ $$1 - 30 T + 502 T^{2} - 5646 T^{3} + 46995 T^{4} - 297598 T^{5} + 1456845 T^{6} - 5425806 T^{7} + 14955082 T^{8} - 27705630 T^{9} + 28629151 T^{10}$$
$37$ $$1 - 4 T + 109 T^{2} - 216 T^{3} + 5126 T^{4} - 5064 T^{5} + 189662 T^{6} - 295704 T^{7} + 5521177 T^{8} - 7496644 T^{9} + 69343957 T^{10}$$
$41$ $$1 - 6 T + 176 T^{2} - 838 T^{3} + 13295 T^{4} - 49000 T^{5} + 545095 T^{6} - 1408678 T^{7} + 12130096 T^{8} - 16954566 T^{9} + 115856201 T^{10}$$
$43$ $$1 + 12 T + 191 T^{2} + 1696 T^{3} + 16226 T^{4} + 101736 T^{5} + 697718 T^{6} + 3135904 T^{7} + 15185837 T^{8} + 41025612 T^{9} + 147008443 T^{10}$$
$47$ $$1 - 10 T + 110 T^{2} + 38 T^{3} - 3609 T^{4} + 58894 T^{5} - 169623 T^{6} + 83942 T^{7} + 11420530 T^{8} - 48796810 T^{9} + 229345007 T^{10}$$
$53$ $$1 - 16 T + 317 T^{2} - 3240 T^{3} + 35942 T^{4} - 254032 T^{5} + 1904926 T^{6} - 9101160 T^{7} + 47194009 T^{8} - 126247696 T^{9} + 418195493 T^{10}$$
$59$ $$1 - 22 T + 413 T^{2} - 5194 T^{3} + 54600 T^{4} - 458288 T^{5} + 3221400 T^{6} - 18080314 T^{7} + 84821527 T^{8} - 266581942 T^{9} + 714924299 T^{10}$$
$61$ $$1 + 18 T + 339 T^{2} + 3954 T^{3} + 43744 T^{4} + 348488 T^{5} + 2668384 T^{6} + 14712834 T^{7} + 76946559 T^{8} + 249225138 T^{9} + 844596301 T^{10}$$
$67$ $$1 + 2 T + 35 T^{2} - 204 T^{3} + 6790 T^{4} + 16644 T^{5} + 454930 T^{6} - 915756 T^{7} + 10526705 T^{8} + 40302242 T^{9} + 1350125107 T^{10}$$
$71$ $$1 - 4 T + 254 T^{2} - 860 T^{3} + 30833 T^{4} - 86976 T^{5} + 2189143 T^{6} - 4335260 T^{7} + 90909394 T^{8} - 101646724 T^{9} + 1804229351 T^{10}$$
$73$ $$1 + 2 T + 168 T^{2} + 946 T^{3} + 18175 T^{4} + 89144 T^{5} + 1326775 T^{6} + 5041234 T^{7} + 65354856 T^{8} + 56796482 T^{9} + 2073071593 T^{10}$$
$79$ $$1 - 30 T + 703 T^{2} - 10676 T^{3} + 136374 T^{4} - 1310412 T^{5} + 10773546 T^{6} - 66628916 T^{7} + 346606417 T^{8} - 1168502430 T^{9} + 3077056399 T^{10}$$
$83$ $$1 - 8 T + 359 T^{2} - 2224 T^{3} + 55778 T^{4} - 264336 T^{5} + 4629574 T^{6} - 15321136 T^{7} + 205271533 T^{8} - 379666568 T^{9} + 3939040643 T^{10}$$
$89$ $$1 + 20 T + 511 T^{2} + 6422 T^{3} + 92672 T^{4} + 821572 T^{5} + 8247808 T^{6} + 50868662 T^{7} + 360239159 T^{8} + 1254844820 T^{9} + 5584059449 T^{10}$$
$97$ $$1 + 12 T + 371 T^{2} + 3294 T^{3} + 61432 T^{4} + 417340 T^{5} + 5958904 T^{6} + 30993246 T^{7} + 338601683 T^{8} + 1062351372 T^{9} + 8587340257 T^{10}$$