# Properties

 Label 931.2.a.o Level $931$ Weight $2$ Character orbit 931.a Self dual yes Analytic conductor $7.434$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$931 = 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 931.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.43407242818$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 6 x^{5} + 18 x^{4} + 4 x^{3} - 12 x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 133) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 6 x^{5} + 18 x^{4} + 4 x^{3} - 12 x^{2} - x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} - 5 \nu^{4} + 16 \nu^{3} - \nu^{2} - 3 \nu - 1$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 3 \nu^{2} + \nu - 3$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 4 \nu^{5} - 3 \nu^{4} + 23 \nu^{3} - 12 \nu^{2} - 11 \nu + 4$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{5} + 9 \nu^{4} - 11 \nu^{3} - 22 \nu^{2} + \nu + 8$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{5} + 9 \nu^{4} - 13 \nu^{3} - 20 \nu^{2} + 15 \nu + 4$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} + 6 \nu^{4} - 18 \nu^{3} - 4 \nu^{2} + 11 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} - \beta_{3} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_{1} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$($$$$-11 \beta_{6} + 9 \beta_{5} + 4 \beta_{4} - 13 \beta_{3} + 14 \beta_{2} + 18 \beta_{1} + 33$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-59 \beta_{6} + 41 \beta_{5} + 22 \beta_{4} - 67 \beta_{3} + 20 \beta_{2} + 32 \beta_{1} + 69$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-54 \beta_{6} + 38 \beta_{5} + 27 \beta_{4} - 71 \beta_{3} + 50 \beta_{2} + 80 \beta_{1} + 127$$

## 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
 −0.770405 0.862998 3.00704 0.273704 2.27137 −2.29398 −0.350729
−2.46342 3.03216 4.06842 0.527614 −7.46948 0 −5.09539 6.19401 −1.29973
1.2 −1.40650 −3.19504 −0.0217491 −0.295752 4.49383 0 2.84360 7.20828 0.415977
1.3 −0.812652 −0.415454 −1.33960 2.67449 0.337619 0 2.71393 −2.82740 −2.17343
1.4 0.269662 −1.39869 −1.92728 −3.37987 −0.377172 0 −1.05904 −1.04367 −0.911422
1.5 1.13506 2.19453 −0.711632 1.83111 2.49093 0 −3.07787 1.81594 2.07842
1.6 2.59421 2.89925 4.72991 −1.85806 7.52126 0 7.08194 5.40568 −4.82019
1.7 2.68364 −1.11676 5.20193 2.50047 −2.99699 0 8.59284 −1.75284 6.71038
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$19$$ $$1$$

## 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 931.2.a.o 7
3.b odd 2 1 8379.2.a.ck 7
7.b odd 2 1 931.2.a.n 7
7.c even 3 2 931.2.f.p 14
7.d odd 6 2 133.2.f.d 14
21.c even 2 1 8379.2.a.cl 7
21.g even 6 2 1197.2.j.l 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.d 14 7.d odd 6 2
931.2.a.n 7 7.b odd 2 1
931.2.a.o 7 1.a even 1 1 trivial
931.2.f.p 14 7.c even 3 2
1197.2.j.l 14 21.g even 6 2
8379.2.a.ck 7 3.b odd 2 1
8379.2.a.cl 7 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(931))$$:

 $$T_{2}^{7} - 2 T_{2}^{6} - 10 T_{2}^{5} + 16 T_{2}^{4} + 27 T_{2}^{3} - 24 T_{2}^{2} - 18 T_{2} + 6$$ $$T_{3}^{7} - 2 T_{3}^{6} - 16 T_{3}^{5} + 26 T_{3}^{4} + 72 T_{3}^{3} - 52 T_{3}^{2} - 128 T_{3} - 40$$ $$T_{5}^{7} - 2 T_{5}^{6} - 14 T_{5}^{5} + 32 T_{5}^{4} + 33 T_{5}^{3} - 90 T_{5}^{2} + 12 T_{5} + 12$$ $$T_{13}^{7} + 6 T_{13}^{6} - 24 T_{13}^{5} - 210 T_{13}^{4} - 104 T_{13}^{3} + 1620 T_{13}^{2} + 3200 T_{13} + 1208$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6 - 18 T - 24 T^{2} + 27 T^{3} + 16 T^{4} - 10 T^{5} - 2 T^{6} + T^{7}$$
$3$ $$-40 - 128 T - 52 T^{2} + 72 T^{3} + 26 T^{4} - 16 T^{5} - 2 T^{6} + T^{7}$$
$5$ $$12 + 12 T - 90 T^{2} + 33 T^{3} + 32 T^{4} - 14 T^{5} - 2 T^{6} + T^{7}$$
$7$ $$T^{7}$$
$11$ $$-135 - 567 T - 549 T^{2} + 51 T^{3} + 131 T^{4} - 13 T^{5} - 7 T^{6} + T^{7}$$
$13$ $$1208 + 3200 T + 1620 T^{2} - 104 T^{3} - 210 T^{4} - 24 T^{5} + 6 T^{6} + T^{7}$$
$17$ $$48 + 336 T + 828 T^{2} + 912 T^{3} + 497 T^{4} + 139 T^{5} + 19 T^{6} + T^{7}$$
$19$ $$( 1 + T )^{7}$$
$23$ $$3165 - 183 T - 3429 T^{2} + 1659 T^{3} + 7 T^{4} - 85 T^{5} + T^{6} + T^{7}$$
$29$ $$12480 - 17184 T + 7536 T^{2} - 348 T^{3} - 668 T^{4} + 206 T^{5} - 24 T^{6} + T^{7}$$
$31$ $$-8632 - 14512 T + 204 T^{2} + 2296 T^{3} - 6 T^{4} - 96 T^{5} + T^{7}$$
$37$ $$155776 - 62208 T - 31744 T^{2} + 5252 T^{3} + 1180 T^{4} - 154 T^{5} - 8 T^{6} + T^{7}$$
$41$ $$768 - 1152 T - 768 T^{2} + 432 T^{3} + 128 T^{4} - 40 T^{5} - 4 T^{6} + T^{7}$$
$43$ $$303284 - 99252 T - 23308 T^{2} + 6787 T^{3} + 556 T^{4} - 148 T^{5} - 4 T^{6} + T^{7}$$
$47$ $$56193 - 54807 T + 4791 T^{2} + 5595 T^{3} - 413 T^{4} - 149 T^{5} + 5 T^{6} + T^{7}$$
$53$ $$-408 + 3264 T + 9300 T^{2} - 6648 T^{3} + 1006 T^{4} + 56 T^{5} - 20 T^{6} + T^{7}$$
$59$ $$-124032 + 68544 T + 288 T^{2} - 6552 T^{3} + 1234 T^{4} - 2 T^{5} - 16 T^{6} + T^{7}$$
$61$ $$-1281545 - 224009 T + 62887 T^{2} + 11331 T^{3} - 995 T^{4} - 187 T^{5} + 5 T^{6} + T^{7}$$
$67$ $$-8192 - 86016 T + 81920 T^{2} + 16448 T^{3} - 1184 T^{4} - 256 T^{5} + 4 T^{6} + T^{7}$$
$71$ $$6648 - 15120 T - 35676 T^{2} + 19416 T^{3} + 2014 T^{4} - 256 T^{5} - 12 T^{6} + T^{7}$$
$73$ $$-164299 + 48035 T + 92889 T^{2} + 19435 T^{3} - 945 T^{4} - 303 T^{5} + 3 T^{6} + T^{7}$$
$79$ $$-151672 + 178416 T + 64652 T^{2} - 6632 T^{3} - 2774 T^{4} - 64 T^{5} + 20 T^{6} + T^{7}$$
$83$ $$-142371 - 203715 T - 8721 T^{2} + 15507 T^{3} + 1295 T^{4} - 209 T^{5} - 11 T^{6} + T^{7}$$
$89$ $$-10464 + 13200 T - 3432 T^{2} - 1020 T^{3} + 488 T^{4} - 22 T^{5} - 10 T^{6} + T^{7}$$
$97$ $$-32 + 608 T - 2784 T^{2} + 3168 T^{3} + 642 T^{4} - 258 T^{5} - 4 T^{6} + T^{7}$$