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

Downloads

Learn more about

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: \(\Q(\sqrt{193}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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.

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

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:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)

Hecke kernels

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