# Properties

 Label 1710.2.t.b Level $1710$ Weight $2$ Character orbit 1710.t Analytic conductor $13.654$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1710.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.6544187456$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.89539436150784.1 Defining polynomial: $$x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} + 4 x^{8} + 16 x^{7} - 8 x^{6} + 20 x^{5} + 20 x^{4} - 24 x^{3} + 8 x^{2} - 8 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 570) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + \beta_{8} q^{4} + ( \beta_{6} - \beta_{11} ) q^{5} + ( 3 \beta_{4} - \beta_{9} ) q^{7} -\beta_{4} q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + \beta_{8} q^{4} + ( \beta_{6} - \beta_{11} ) q^{5} + ( 3 \beta_{4} - \beta_{9} ) q^{7} -\beta_{4} q^{8} + ( \beta_{2} + \beta_{5} + \beta_{9} + \beta_{10} ) q^{10} + ( 1 + 2 \beta_{2} ) q^{11} + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{13} + ( 3 - 3 \beta_{8} + \beta_{11} ) q^{14} + ( -1 + \beta_{8} ) q^{16} + ( \beta_{5} - 2 \beta_{6} ) q^{17} + ( \beta_{1} - \beta_{2} + \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{19} + ( -\beta_{1} - \beta_{7} ) q^{20} + ( \beta_{3} + 2 \beta_{6} ) q^{22} + ( 4 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 3 \beta_{9} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{25} + ( 1 - \beta_{1} + \beta_{2} ) q^{26} + ( 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{9} ) q^{28} + ( 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{8} - 2 \beta_{10} - 3 \beta_{11} ) q^{29} + 2 \beta_{1} q^{31} + ( -\beta_{3} - \beta_{4} ) q^{32} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{10} + \beta_{11} ) q^{34} + ( 1 + 3 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{8} - 4 \beta_{10} ) q^{35} + ( -2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{37} + ( -\beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{38} + ( \beta_{5} + \beta_{10} ) q^{40} + ( -5 \beta_{10} - 2 \beta_{11} ) q^{41} + ( -4 \beta_{3} + 2 \beta_{5} ) q^{43} + ( 2 \beta_{2} + \beta_{8} + 2 \beta_{10} ) q^{44} + ( 4 - 3 \beta_{1} + 5 \beta_{2} ) q^{46} + ( 7 \beta_{3} + 7 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 3 \beta_{9} ) q^{47} + ( -5 - 5 \beta_{1} - \beta_{2} ) q^{49} + ( -2 - 2 \beta_{2} + \beta_{4} + 2 \beta_{7} + 2 \beta_{9} ) q^{50} + ( \beta_{3} + \beta_{5} + \beta_{6} ) q^{52} + ( 6 \beta_{3} + 6 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{53} + ( -2 + 4 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{55} + ( 3 + \beta_{1} ) q^{56} + ( -2 \beta_{4} + 2 \beta_{7} + 3 \beta_{9} ) q^{58} + ( -1 + \beta_{8} - \beta_{10} - 5 \beta_{11} ) q^{59} + ( 5 \beta_{2} - 3 \beta_{8} + 5 \beta_{10} ) q^{61} -2 \beta_{5} q^{62} - q^{64} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{9} ) q^{65} + ( 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{9} ) q^{67} + ( 2 \beta_{7} - \beta_{9} ) q^{68} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{70} + ( -7 + 7 \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{71} + ( 7 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{73} + ( -2 + 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{74} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{76} + ( 5 \beta_{4} + 8 \beta_{7} - \beta_{9} ) q^{77} + ( -6 + 6 \beta_{8} - 2 \beta_{11} ) q^{79} + ( -\beta_{1} - \beta_{6} - \beta_{7} + \beta_{11} ) q^{80} + ( 2 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 2 \beta_{9} ) q^{82} + ( -4 \beta_{4} - 4 \beta_{7} ) q^{83} + ( -2 \beta_{1} + \beta_{2} + 5 \beta_{3} + 5 \beta_{4} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{8} + 2 \beta_{11} ) q^{86} + ( -\beta_{4} - 2 \beta_{7} ) q^{88} + ( 5 \beta_{1} + 6 \beta_{2} + \beta_{8} + 6 \beta_{10} - 5 \beta_{11} ) q^{89} + ( \beta_{1} - 3 \beta_{2} - \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{91} + ( 4 \beta_{3} + 3 \beta_{5} + 5 \beta_{6} ) q^{92} + ( 7 - 3 \beta_{1} + 5 \beta_{2} ) q^{94} + ( -2 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{95} + ( -5 \beta_{3} - 5 \beta_{6} ) q^{97} + ( -5 \beta_{3} + 5 \beta_{5} - \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 6q^{4} + O(q^{10})$$ $$12q + 6q^{4} - 2q^{10} + 4q^{11} + 18q^{14} - 6q^{16} + 6q^{19} - 2q^{25} + 8q^{26} + 16q^{29} + 4q^{34} - 2q^{35} + 2q^{40} - 10q^{41} + 2q^{44} + 28q^{46} - 56q^{49} - 16q^{50} - 8q^{55} + 36q^{56} - 8q^{59} - 28q^{61} - 12q^{64} + 16q^{65} + 16q^{70} - 44q^{71} - 14q^{74} - 12q^{76} - 36q^{79} - 32q^{85} - 24q^{86} - 6q^{89} + 64q^{94} + 12q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} + 4 x^{8} + 16 x^{7} - 8 x^{6} + 20 x^{5} + 20 x^{4} - 24 x^{3} + 8 x^{2} - 8 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + 144 \nu^{3} - 48 \nu^{2} + 48 \nu + 748$$$$)/460$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} - 454 \nu^{4} + 444 \nu^{3} - 148 \nu^{2} + 148 \nu + 612$$$$)/460$$ $$\beta_{3}$$ $$=$$ $$($$$$12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + 110 \nu^{4} - 50 \nu^{3} - 1370 \nu^{2} + 12 \nu - 12$$$$)/460$$ $$\beta_{4}$$ $$=$$ $$($$$$31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142$$$$)/230$$ $$\beta_{5}$$ $$=$$ $$($$$$-12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} - 110 \nu^{4} + 50 \nu^{3} + 1140 \nu^{2} - 12 \nu + 12$$$$)/230$$ $$\beta_{6}$$ $$=$$ $$($$$$32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + 278 \nu^{4} - 118 \nu^{3} - 1982 \nu^{2} + 32 \nu - 32$$$$)/460$$ $$\beta_{7}$$ $$=$$ $$($$$$81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + 1614 \nu^{4} + 3424 \nu^{3} + 484 \nu^{2} - 484 \nu - 420$$$$)/460$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12$$$$)/20$$ $$\beta_{9}$$ $$=$$ $$($$$$-101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} - 1774 \nu^{4} - 3856 \nu^{3} - 524 \nu^{2} + 524 \nu + 476$$$$)/460$$ $$\beta_{10}$$ $$=$$ $$($$$$117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + 2510 \nu^{4} + 2790 \nu^{3} - 1294 \nu^{2} + 1060 \nu - 1060$$$$)/460$$ $$\beta_{11}$$ $$=$$ $$($$$$-60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} - 1424 \nu^{4} - 1636 \nu^{3} + 732 \nu^{2} - 612 \nu + 612$$$$)/230$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} - 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{9} + \beta_{7} + 2 \beta_{4} + \beta_{2} - 2 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{11} + \beta_{10} + 7 \beta_{8} + \beta_{2} - 5 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{11} + 3 \beta_{10} + 9 \beta_{8} - 3 \beta_{6} - 8 \beta_{5} - 9 \beta_{3} - 9$$ $$\nu^{6}$$ $$=$$ $$22 \beta_{9} + 6 \beta_{7} + 28 \beta_{4}$$ $$\nu^{7}$$ $$=$$ $$33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 33 \beta_{5} + 39 \beta_{4} + 39 \beta_{3} + 11 \beta_{2} - 33 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$94 \beta_{11} + 28 \beta_{10} + 116 \beta_{8} - 116$$ $$\nu^{9}$$ $$=$$ $$138 \beta_{9} + 44 \beta_{7} + 166 \beta_{4} - 44 \beta_{2} + 138 \beta_{1} - 166$$ $$\nu^{10}$$ $$=$$ $$398 \beta_{9} + 122 \beta_{7} + 122 \beta_{6} + 398 \beta_{5} + 486 \beta_{4} + 486 \beta_{3}$$ $$\nu^{11}$$ $$=$$ $$580 \beta_{11} + 182 \beta_{10} + 702 \beta_{8} + 182 \beta_{6} + 580 \beta_{5} + 702 \beta_{3} - 702$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
919.1
 0.550552 − 0.147520i 1.98293 − 0.531325i −1.16746 + 0.312819i 0.312819 + 1.16746i −0.147520 − 0.550552i −0.531325 − 1.98293i 0.550552 + 0.147520i 1.98293 + 0.531325i −1.16746 − 0.312819i 0.312819 − 1.16746i −0.147520 + 0.550552i −0.531325 + 1.98293i
−0.866025 + 0.500000i 0 0.500000 0.866025i −2.12032 0.710109i 0 4.67513i 1.00000i 0 2.19130 0.445186i
919.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.37659 + 1.76210i 0 0.785680i 1.00000i 0 −2.07321 0.837733i
919.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.60976 1.55199i 0 3.53919i 1.00000i 0 −0.618092 + 2.14894i
919.4 0.866025 0.500000i 0 0.500000 0.866025i −2.14894 + 0.618092i 0 3.53919i 1.00000i 0 −1.55199 + 1.60976i
919.5 0.866025 0.500000i 0 0.500000 0.866025i 0.445186 2.19130i 0 4.67513i 1.00000i 0 −0.710109 2.12032i
919.6 0.866025 0.500000i 0 0.500000 0.866025i 0.837733 + 2.07321i 0 0.785680i 1.00000i 0 1.76210 + 1.37659i
1189.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.12032 + 0.710109i 0 4.67513i 1.00000i 0 2.19130 + 0.445186i
1189.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.37659 1.76210i 0 0.785680i 1.00000i 0 −2.07321 + 0.837733i
1189.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.60976 + 1.55199i 0 3.53919i 1.00000i 0 −0.618092 2.14894i
1189.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.14894 0.618092i 0 3.53919i 1.00000i 0 −1.55199 1.60976i
1189.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.445186 + 2.19130i 0 4.67513i 1.00000i 0 −0.710109 + 2.12032i
1189.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.837733 2.07321i 0 0.785680i 1.00000i 0 1.76210 1.37659i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1189.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.t.b 12
3.b odd 2 1 570.2.q.b 12
5.b even 2 1 inner 1710.2.t.b 12
15.d odd 2 1 570.2.q.b 12
19.c even 3 1 inner 1710.2.t.b 12
57.h odd 6 1 570.2.q.b 12
95.i even 6 1 inner 1710.2.t.b 12
285.n odd 6 1 570.2.q.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.q.b 12 3.b odd 2 1
570.2.q.b 12 15.d odd 2 1
570.2.q.b 12 57.h odd 6 1
570.2.q.b 12 285.n odd 6 1
1710.2.t.b 12 1.a even 1 1 trivial
1710.2.t.b 12 5.b even 2 1 inner
1710.2.t.b 12 19.c even 3 1 inner
1710.2.t.b 12 95.i even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} + 35 T_{7}^{4} + 295 T_{7}^{2} + 169$$ acting on $$S_{2}^{\mathrm{new}}(1710, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{3}$$
$3$ $$T^{12}$$
$5$ $$15625 + 625 T^{2} + 4000 T^{3} + 150 T^{4} + 80 T^{5} + 501 T^{6} + 16 T^{7} + 6 T^{8} + 32 T^{9} + T^{10} + T^{12}$$
$7$ $$( 169 + 295 T^{2} + 35 T^{4} + T^{6} )^{2}$$
$11$ $$( 5 - 13 T - T^{2} + T^{3} )^{4}$$
$13$ $$256 - 512 T^{2} + 832 T^{4} - 352 T^{6} + 112 T^{8} - 12 T^{10} + T^{12}$$
$17$ $$6250000 - 1500000 T^{2} + 250000 T^{4} - 21400 T^{6} + 1336 T^{8} - 44 T^{10} + T^{12}$$
$19$ $$( 6859 - 1083 T + 456 T^{2} - 25 T^{3} + 24 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$23$ $$58120048561 - 2941911443 T^{2} + 101902414 T^{4} - 1897423 T^{6} + 25822 T^{8} - 195 T^{10} + T^{12}$$
$29$ $$( 16900 - 2080 T + 1296 T^{2} - 132 T^{3} + 80 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$31$ $$( 16 - 16 T + T^{3} )^{4}$$
$37$ $$( 1 + 107 T^{2} + 27 T^{4} + T^{6} )^{2}$$
$41$ $$( 19321 - 9869 T + 4346 T^{2} - 633 T^{3} + 96 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$43$ $$65536 - 360448 T^{2} + 1961984 T^{4} - 112128 T^{6} + 4992 T^{8} - 80 T^{10} + T^{12}$$
$47$ $$147763360000 - 7632646400 T^{2} + 292779136 T^{4} - 4473184 T^{6} + 49840 T^{8} - 264 T^{10} + T^{12}$$
$53$ $$332150625 - 146146275 T^{2} + 60531786 T^{4} - 1623483 T^{6} + 34830 T^{8} - 207 T^{10} + T^{12}$$
$59$ $$( 204304 + 39776 T + 9552 T^{2} + 552 T^{3} + 104 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$61$ $$( 77284 + 5004 T + 4216 T^{2} + 304 T^{3} + 214 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$67$ $$102627966736 - 4561869440 T^{2} + 136143552 T^{4} - 2321208 T^{6} + 29024 T^{8} - 208 T^{10} + T^{12}$$
$71$ $$( 42436 + 28428 T + 14512 T^{2} + 2624 T^{3} + 346 T^{4} + 22 T^{5} + T^{6} )^{2}$$
$73$ $$623201296 - 144291920 T^{2} + 28914880 T^{4} - 990472 T^{6} + 26620 T^{8} - 180 T^{10} + T^{12}$$
$79$ $$( 10816 + 9568 T + 6592 T^{2} + 1448 T^{3} + 232 T^{4} + 18 T^{5} + T^{6} )^{2}$$
$83$ $$( 16384 + 3072 T^{2} + 128 T^{4} + T^{6} )^{2}$$
$89$ $$( 4133089 + 563141 T + 82828 T^{2} + 3235 T^{3} + 286 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$97$ $$3906250000 - 468750000 T^{2} + 43750000 T^{4} - 1375000 T^{6} + 32500 T^{8} - 200 T^{10} + T^{12}$$