# Properties

 Label 7.10.a.a Level $7$ Weight $10$ Character orbit 7.a Self dual yes Analytic conductor $3.605$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 7.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.60525085315$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{193}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 - \beta ) q^{2} + ( -43 + 11 \beta ) q^{3} + ( -310 + 6 \beta ) q^{4} + ( -1119 - 95 \beta ) q^{5} + ( -1994 + 10 \beta ) q^{6} -2401 q^{7} + ( 1308 + 804 \beta ) q^{8} + ( 5519 - 946 \beta ) q^{9} +O(q^{10})$$ $$q + ( -3 - \beta ) q^{2} + ( -43 + 11 \beta ) q^{3} + ( -310 + 6 \beta ) q^{4} + ( -1119 - 95 \beta ) q^{5} + ( -1994 + 10 \beta ) q^{6} -2401 q^{7} + ( 1308 + 804 \beta ) q^{8} + ( 5519 - 946 \beta ) q^{9} + ( 21692 + 1404 \beta ) q^{10} + ( 17658 - 3326 \beta ) q^{11} + ( 26068 - 3668 \beta ) q^{12} + ( -13265 + 10899 \beta ) q^{13} + ( 7203 + 2401 \beta ) q^{14} + ( -153568 - 8224 \beta ) q^{15} + ( -376 - 6792 \beta ) q^{16} + ( -231960 + 9426 \beta ) q^{17} + ( 166021 - 2681 \beta ) q^{18} + ( -462713 - 1887 \beta ) q^{19} + ( 236880 + 22736 \beta ) q^{20} + ( 103243 - 26411 \beta ) q^{21} + ( 588944 - 7680 \beta ) q^{22} + ( 389064 - 38088 \beta ) q^{23} + ( 1650648 - 20184 \beta ) q^{24} + ( 1040861 + 212610 \beta ) q^{25} + ( -2063712 - 19432 \beta ) q^{26} + ( -1399306 - 115126 \beta ) q^{27} + ( 744310 - 14406 \beta ) q^{28} + ( -5001792 - 94682 \beta ) q^{29} + ( 2047936 + 178240 \beta ) q^{30} + ( 1233630 - 161430 \beta ) q^{31} + ( 642288 - 390896 \beta ) q^{32} + ( -7820392 + 337256 \beta ) q^{33} + ( -1123338 + 203682 \beta ) q^{34} + ( 2686719 + 228095 \beta ) q^{35} + ( -2806358 + 326374 \beta ) q^{36} + ( 15367776 - 248130 \beta ) q^{37} + ( 1752330 + 468374 \beta ) q^{38} + ( 23708972 - 614572 \beta ) q^{39} + ( -16204992 - 1023936 \beta ) q^{40} + ( -9551724 - 860818 \beta ) q^{41} + ( 4787594 - 24010 \beta ) q^{42} + ( 2032550 + 1048278 \beta ) q^{43} + ( -9325488 + 1137008 \beta ) q^{44} + ( 11169149 + 534269 \beta ) q^{45} + ( 6183792 - 274800 \beta ) q^{46} + ( -41097510 + 1033182 \beta ) q^{47} + ( -14403248 + 287920 \beta ) q^{48} + 5764801 q^{49} + ( -44156313 - 1678691 \beta ) q^{50} + ( 29985678 - 2956878 \beta ) q^{51} + ( 16733192 - 3458280 \beta ) q^{52} + ( -27594906 + 4685568 \beta ) q^{53} + ( 26417236 + 1744684 \beta ) q^{54} + ( 41222908 + 2044284 \beta ) q^{55} + ( -3140508 - 1930404 \beta ) q^{56} + ( 15890558 - 5008702 \beta ) q^{57} + ( 33279002 + 5285838 \beta ) q^{58} + ( -3534609 + 1563825 \beta ) q^{59} + ( 38082688 + 1628032 \beta ) q^{60} + ( 22158193 - 3395319 \beta ) q^{61} + ( 27455100 - 749340 \beta ) q^{62} + ( -13251119 + 2271346 \beta ) q^{63} + ( 73708576 + 4007904 \beta ) q^{64} + ( -184989630 - 10935806 \beta ) q^{65} + ( -41629232 + 6808624 \beta ) q^{66} + ( -120960668 - 7026216 \beta ) q^{67} + ( 82822908 - 4313820 \beta ) q^{68} + ( -97590576 + 5917488 \beta ) q^{69} + ( -52082492 - 3371004 \beta ) q^{70} + ( 103246908 + 15075900 \beta ) q^{71} + ( -139573860 + 3199908 \beta ) q^{72} + ( -249576594 - 2840484 \beta ) q^{73} + ( 1785762 - 14623386 \beta ) q^{74} + ( 406614007 + 2307241 \beta ) q^{75} + ( 141255884 - 2191308 \beta ) q^{76} + ( -42396858 + 7985726 \beta ) q^{77} + ( 47485480 - 21865256 \beta ) q^{78} + ( 234267548 + 16873716 \beta ) q^{79} + ( 124952064 + 7635968 \beta ) q^{80} + ( -292872817 + 8178170 \beta ) q^{81} + ( 194793046 + 12134178 \beta ) q^{82} + ( 222011979 - 21562275 \beta ) q^{83} + ( -62589268 + 8806868 \beta ) q^{84} + ( 86737530 + 11488506 \beta ) q^{85} + ( -208415304 - 5177384 \beta ) q^{86} + ( 14067170 - 50948386 \beta ) q^{87} + ( -493005408 + 9846624 \beta ) q^{88} + ( 318133698 + 1406968 \beta ) q^{89} + ( -136621364 - 12771956 \beta ) q^{90} + ( 31849265 - 26168499 \beta ) q^{91} + ( -164715744 + 14141664 \beta ) q^{92} + ( -395761980 + 20511420 \beta ) q^{93} + ( -76111596 + 37997964 \beta ) q^{94} + ( 552373992 + 46069288 \beta ) q^{95} + ( -857490592 + 23873696 \beta ) q^{96} + ( -816358032 + 5731530 \beta ) q^{97} + ( -17294403 - 5764801 \beta ) q^{98} + ( 704708930 - 35060662 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{2} - 86 q^{3} - 620 q^{4} - 2238 q^{5} - 3988 q^{6} - 4802 q^{7} + 2616 q^{8} + 11038 q^{9} + O(q^{10})$$ $$2 q - 6 q^{2} - 86 q^{3} - 620 q^{4} - 2238 q^{5} - 3988 q^{6} - 4802 q^{7} + 2616 q^{8} + 11038 q^{9} + 43384 q^{10} + 35316 q^{11} + 52136 q^{12} - 26530 q^{13} + 14406 q^{14} - 307136 q^{15} - 752 q^{16} - 463920 q^{17} + 332042 q^{18} - 925426 q^{19} + 473760 q^{20} + 206486 q^{21} + 1177888 q^{22} + 778128 q^{23} + 3301296 q^{24} + 2081722 q^{25} - 4127424 q^{26} - 2798612 q^{27} + 1488620 q^{28} - 10003584 q^{29} + 4095872 q^{30} + 2467260 q^{31} + 1284576 q^{32} - 15640784 q^{33} - 2246676 q^{34} + 5373438 q^{35} - 5612716 q^{36} + 30735552 q^{37} + 3504660 q^{38} + 47417944 q^{39} - 32409984 q^{40} - 19103448 q^{41} + 9575188 q^{42} + 4065100 q^{43} - 18650976 q^{44} + 22338298 q^{45} + 12367584 q^{46} - 82195020 q^{47} - 28806496 q^{48} + 11529602 q^{49} - 88312626 q^{50} + 59971356 q^{51} + 33466384 q^{52} - 55189812 q^{53} + 52834472 q^{54} + 82445816 q^{55} - 6281016 q^{56} + 31781116 q^{57} + 66558004 q^{58} - 7069218 q^{59} + 76165376 q^{60} + 44316386 q^{61} + 54910200 q^{62} - 26502238 q^{63} + 147417152 q^{64} - 369979260 q^{65} - 83258464 q^{66} - 241921336 q^{67} + 165645816 q^{68} - 195181152 q^{69} - 104164984 q^{70} + 206493816 q^{71} - 279147720 q^{72} - 499153188 q^{73} + 3571524 q^{74} + 813228014 q^{75} + 282511768 q^{76} - 84793716 q^{77} + 94970960 q^{78} + 468535096 q^{79} + 249904128 q^{80} - 585745634 q^{81} + 389586092 q^{82} + 444023958 q^{83} - 125178536 q^{84} + 173475060 q^{85} - 416830608 q^{86} + 28134340 q^{87} - 986010816 q^{88} + 636267396 q^{89} - 273242728 q^{90} + 63698530 q^{91} - 329431488 q^{92} - 791523960 q^{93} - 152223192 q^{94} + 1104747984 q^{95} - 1714981184 q^{96} - 1632716064 q^{97} - 34588806 q^{98} + 1409417860 q^{99} + O(q^{100})$$

## 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
 7.44622 −6.44622
−16.8924 109.817 −226.645 −2438.78 −1855.08 −2401.00 12477.5 −7623.25 41197.0
1.2 10.8924 −195.817 −393.355 200.782 −2132.92 −2401.00 −9861.52 18661.3 2187.01
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$

## 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 7.10.a.a 2
3.b odd 2 1 63.10.a.d 2
4.b odd 2 1 112.10.a.e 2
5.b even 2 1 175.10.a.b 2
5.c odd 4 2 175.10.b.b 4
7.b odd 2 1 49.10.a.b 2
7.c even 3 2 49.10.c.c 4
7.d odd 6 2 49.10.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.a 2 1.a even 1 1 trivial
49.10.a.b 2 7.b odd 2 1
49.10.c.b 4 7.d odd 6 2
49.10.c.c 4 7.c even 3 2
63.10.a.d 2 3.b odd 2 1
112.10.a.e 2 4.b odd 2 1
175.10.a.b 2 5.b even 2 1
175.10.b.b 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 6 T_{2} - 184$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(7))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-184 + 6 T + T^{2}$$
$3$ $$-21504 + 86 T + T^{2}$$
$5$ $$-489664 + 2238 T + T^{2}$$
$7$ $$( 2401 + T )^{2}$$
$11$ $$-1823214304 - 35316 T + T^{2}$$
$13$ $$-22750162568 + 26530 T + T^{2}$$
$17$ $$36657492732 + 463920 T + T^{2}$$
$19$ $$213416091952 + 925426 T + T^{2}$$
$23$ $$-128613482496 - 778128 T + T^{2}$$
$29$ $$23287739754332 + 10003584 T + T^{2}$$
$31$ $$-3507668488800 - 2467260 T + T^{2}$$
$37$ $$224285819284476 - 30735552 T + T^{2}$$
$41$ $$-51779041048756 + 19103448 T + T^{2}$$
$43$ $$-207953886197312 - 4065100 T + T^{2}$$
$47$ $$1482984574491168 + 82195020 T + T^{2}$$
$53$ $$-3475748826997596 + 55189812 T + T^{2}$$
$59$ $$-459497424927744 + 7069218 T + T^{2}$$
$61$ $$-1733955367544624 - 44316386 T + T^{2}$$
$67$ $$5103514926225616 + 241921336 T + T^{2}$$
$71$ $$-33205648824769536 - 206493816 T + T^{2}$$
$73$ $$60731284847269428 + 499153188 T + T^{2}$$
$79$ $$-70118242258304 - 468535096 T + T^{2}$$
$83$ $$-40442499893399184 - 444023958 T + T^{2}$$
$89$ $$100826994925221572 - 636267396 T + T^{2}$$
$97$ $$660100302235719324 + 1632716064 T + T^{2}$$