Properties

Label 7.10.a.b
Level 7
Weight 10
Character orbit 7.a
Self dual Yes
Analytic conductor 3.605
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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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: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 7 - \beta_{2} ) q^{2} \) \( + ( 28 - \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 519 + 7 \beta_{1} - 8 \beta_{2} ) q^{4} \) \( + ( 518 - 13 \beta_{1} + 43 \beta_{2} ) q^{5} \) \( + ( 1638 - 6 \beta_{1} + 36 \beta_{2} ) q^{6} \) \( + 2401 q^{7} \) \( + ( 4685 + 147 \beta_{1} - 470 \beta_{2} ) q^{8} \) \( + ( -8667 - 126 \beta_{1} + 90 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 7 - \beta_{2} ) q^{2} \) \( + ( 28 - \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 519 + 7 \beta_{1} - 8 \beta_{2} ) q^{4} \) \( + ( 518 - 13 \beta_{1} + 43 \beta_{2} ) q^{5} \) \( + ( 1638 - 6 \beta_{1} + 36 \beta_{2} ) q^{6} \) \( + 2401 q^{7} \) \( + ( 4685 + 147 \beta_{1} - 470 \beta_{2} ) q^{8} \) \( + ( -8667 - 126 \beta_{1} + 90 \beta_{2} ) q^{9} \) \( + ( -32620 - 470 \beta_{1} + 370 \beta_{2} ) q^{10} \) \( + ( -1148 + 658 \beta_{1} + 650 \beta_{2} ) q^{11} \) \( + ( -35462 + 182 \beta_{1} - 700 \beta_{2} ) q^{12} \) \( + ( -6594 - 175 \beta_{1} + 3017 \beta_{2} ) q^{13} \) \( + ( 16807 - 2401 \beta_{2} ) q^{14} \) \( + ( 66768 - 1848 \beta_{1} - 1392 \beta_{2} ) q^{15} \) \( + ( 160987 + 1617 \beta_{1} - 10614 \beta_{2} ) q^{16} \) \( + ( 338898 + 1574 \beta_{1} + 3030 \beta_{2} ) q^{17} \) \( + ( -91089 - 2268 \beta_{1} + 16947 \beta_{2} ) q^{18} \) \( + ( 74284 + 2437 \beta_{1} - 15371 \beta_{2} ) q^{19} \) \( + ( -640696 - 2044 \beta_{1} + 41524 \beta_{2} ) q^{20} \) \( + ( 67228 - 2401 \beta_{1} - 2401 \beta_{2} ) q^{21} \) \( + ( -949016 + 4004 \beta_{1} - 40972 \beta_{2} ) q^{22} \) \( + ( 628544 - 3808 \beta_{1} - 24200 \beta_{2} ) q^{23} \) \( + ( -483210 + 10338 \beta_{1} + 4500 \beta_{2} ) q^{24} \) \( + ( 1024407 - 4802 \beta_{1} - 15338 \beta_{2} ) q^{25} \) \( + ( -2928352 - 23394 \beta_{1} + 20986 \beta_{2} ) q^{26} \) \( + ( 183960 + 15786 \beta_{1} + 33930 \beta_{2} ) q^{27} \) \( + ( 1246119 + 16807 \beta_{1} - 19208 \beta_{2} ) q^{28} \) \( + ( 1360606 + 18914 \beta_{1} + 54866 \beta_{2} ) q^{29} \) \( + ( 2684400 - 14280 \beta_{1} + 51960 \beta_{2} ) q^{30} \) \( + ( 956480 - 70302 \beta_{1} + 55698 \beta_{2} ) q^{31} \) \( + ( 8407317 + 20055 \beta_{1} - 36066 \beta_{2} ) q^{32} \) \( + ( -6753264 + 65640 \beta_{1} - 76152 \beta_{2} ) q^{33} \) \( + ( -1327214 - 748 \beta_{1} - 438178 \beta_{2} ) q^{34} \) \( + ( 1243718 - 31213 \beta_{1} + 103243 \beta_{2} ) q^{35} \) \( + ( -11798793 - 83601 \beta_{1} + 209376 \beta_{2} ) q^{36} \) \( + ( 465206 + 60522 \beta_{1} + 209418 \beta_{2} ) q^{37} \) \( + ( 14493290 + 139278 \beta_{1} - 248060 \beta_{2} ) q^{38} \) \( + ( -2996896 - 13748 \beta_{1} - 110012 \beta_{2} ) q^{39} \) \( + ( -27619760 - 76600 \beta_{1} + 625640 \beta_{2} ) q^{40} \) \( + ( -4806886 - 131894 \beta_{1} - 163478 \beta_{2} ) q^{41} \) \( + ( 3932838 - 14406 \beta_{1} + 86436 \beta_{2} ) q^{42} \) \( + ( -20543724 + 65366 \beta_{1} + 121982 \beta_{2} ) q^{43} \) \( + ( 32337328 + 1960 \beta_{1} + 314984 \beta_{2} ) q^{44} \) \( + ( 9924894 + 7551 \beta_{1} - 714681 \beta_{2} ) q^{45} \) \( + ( 29915888 + 119896 \beta_{1} - 405224 \beta_{2} ) q^{46} \) \( + ( -3456320 + 83238 \beta_{1} - 534778 \beta_{2} ) q^{47} \) \( + ( 5599594 + 9710 \beta_{1} + 174140 \beta_{2} ) q^{48} \) \( + 5764801 q^{49} \) \( + ( 24441685 + 44940 \beta_{1} - 727615 \beta_{2} ) q^{50} \) \( + ( -8715576 - 186102 \beta_{1} - 587238 \beta_{2} ) q^{51} \) \( + ( -26969348 - 361424 \beta_{1} + 2925244 \beta_{2} ) q^{52} \) \( + ( 22500870 - 450352 \beta_{1} + 1553376 \beta_{2} ) q^{53} \) \( + ( -39293100 - 32292 \beta_{1} - 1176120 \beta_{2} ) q^{54} \) \( + ( -35274344 + 988364 \beta_{1} + 1659356 \beta_{2} ) q^{55} \) \( + ( 11248685 + 352947 \beta_{1} - 1128470 \beta_{2} ) q^{56} \) \( + ( 2823704 + 182350 \beta_{1} + 403894 \beta_{2} ) q^{57} \) \( + ( -53054610 - 138180 \beta_{1} - 2535150 \beta_{2} ) q^{58} \) \( + ( -14196700 - 49659 \beta_{1} - 2231195 \beta_{2} ) q^{59} \) \( + ( -59850336 + 396816 \beta_{1} - 991536 \beta_{2} ) q^{60} \) \( + ( 63915614 - 844773 \beta_{1} + 589107 \beta_{2} ) q^{61} \) \( + ( -15661156 - 1303812 \beta_{1} + 3668848 \beta_{2} ) q^{62} \) \( + ( -20809467 - 302526 \beta_{1} + 216090 \beta_{2} ) q^{63} \) \( + ( 2617387 - 314727 \beta_{1} - 4312590 \beta_{2} ) q^{64} \) \( + ( 121427740 + 1035790 \beta_{1} - 958090 \beta_{2} ) q^{65} \) \( + ( -2685984 + 1386384 \beta_{1} + 2410512 \beta_{2} ) q^{66} \) \( + ( -85058596 + 47712 \beta_{1} + 3939816 \beta_{2} ) q^{67} \) \( + ( 247828602 + 2251634 \beta_{1} - 613704 \beta_{2} ) q^{68} \) \( + ( 85967952 - 981336 \beta_{1} + 697656 \beta_{2} ) q^{69} \) \( + ( -78320620 - 1128470 \beta_{1} + 888370 \beta_{2} ) q^{70} \) \( + ( 98838168 - 526260 \beta_{1} + 3499356 \beta_{2} ) q^{71} \) \( + ( -203104755 - 1391229 \beta_{1} + 8765370 \beta_{2} ) q^{72} \) \( + ( 114737770 + 118516 \beta_{1} - 9544844 \beta_{2} ) q^{73} \) \( + ( -230232154 - 679140 \beta_{1} - 4189718 \beta_{2} ) q^{74} \) \( + ( 93010372 - 1484467 \beta_{1} - 4723 \beta_{2} ) q^{75} \) \( + ( 242946662 + 2299290 \beta_{1} - 15924468 \beta_{2} ) q^{76} \) \( + ( -2756348 + 1579858 \beta_{1} + 1560650 \beta_{2} ) q^{77} \) \( + ( 93377592 + 591360 \beta_{1} + 3780504 \beta_{2} ) q^{78} \) \( + ( -320137552 + 1679412 \beta_{1} - 7475532 \beta_{2} ) q^{79} \) \( + ( -444444448 - 4328752 \beta_{1} + 11964112 \beta_{2} ) q^{80} \) \( + ( -11942559 + 3824982 \beta_{1} - 4598370 \beta_{2} ) q^{81} \) \( + ( 187558434 - 570276 \beta_{1} + 13216518 \beta_{2} ) q^{82} \) \( + ( -366839060 - 1977367 \beta_{1} - 4479559 \beta_{2} ) q^{83} \) \( + ( -85144262 + 436982 \beta_{1} - 1680700 \beta_{2} ) q^{84} \) \( + ( 146059804 - 1518034 \beta_{1} + 18558254 \beta_{2} ) q^{85} \) \( + ( -293660752 - 4116 \beta_{1} + 16416916 \beta_{2} ) q^{86} \) \( + ( -207273864 + 457014 \beta_{1} - 5138394 \beta_{2} ) q^{87} \) \( + ( 402041600 - 4229456 \beta_{1} - 11172080 \beta_{2} ) q^{88} \) \( + ( 168938826 + 1815976 \beta_{1} - 10612104 \beta_{2} ) q^{89} \) \( + ( 767817540 + 5100930 \beta_{1} - 11130390 \beta_{2} ) q^{90} \) \( + ( -15832194 - 420175 \beta_{1} + 7243817 \beta_{2} ) q^{91} \) \( + ( 230374496 + 6344912 \beta_{1} - 25723952 \beta_{2} ) q^{92} \) \( + ( 564419504 - 7972076 \beta_{1} + 1921156 \beta_{2} ) q^{93} \) \( + ( 462668276 + 4825540 \beta_{1} - 2488928 \beta_{2} ) q^{94} \) \( + ( -734357024 - 4098416 \beta_{1} + 10749416 \beta_{2} ) q^{95} \) \( + ( 111128598 - 6385806 \beta_{1} - 8360604 \beta_{2} ) q^{96} \) \( + ( -215832750 - 864850 \beta_{1} + 33276782 \beta_{2} ) q^{97} \) \( + ( 40353607 - 5764801 \beta_{2} ) q^{98} \) \( + ( -633659724 + 376362 \beta_{1} - 7689150 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 21q^{2} \) \(\mathstrut +\mathstrut 84q^{3} \) \(\mathstrut +\mathstrut 1557q^{4} \) \(\mathstrut +\mathstrut 1554q^{5} \) \(\mathstrut +\mathstrut 4914q^{6} \) \(\mathstrut +\mathstrut 7203q^{7} \) \(\mathstrut +\mathstrut 14055q^{8} \) \(\mathstrut -\mathstrut 26001q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 21q^{2} \) \(\mathstrut +\mathstrut 84q^{3} \) \(\mathstrut +\mathstrut 1557q^{4} \) \(\mathstrut +\mathstrut 1554q^{5} \) \(\mathstrut +\mathstrut 4914q^{6} \) \(\mathstrut +\mathstrut 7203q^{7} \) \(\mathstrut +\mathstrut 14055q^{8} \) \(\mathstrut -\mathstrut 26001q^{9} \) \(\mathstrut -\mathstrut 97860q^{10} \) \(\mathstrut -\mathstrut 3444q^{11} \) \(\mathstrut -\mathstrut 106386q^{12} \) \(\mathstrut -\mathstrut 19782q^{13} \) \(\mathstrut +\mathstrut 50421q^{14} \) \(\mathstrut +\mathstrut 200304q^{15} \) \(\mathstrut +\mathstrut 482961q^{16} \) \(\mathstrut +\mathstrut 1016694q^{17} \) \(\mathstrut -\mathstrut 273267q^{18} \) \(\mathstrut +\mathstrut 222852q^{19} \) \(\mathstrut -\mathstrut 1922088q^{20} \) \(\mathstrut +\mathstrut 201684q^{21} \) \(\mathstrut -\mathstrut 2847048q^{22} \) \(\mathstrut +\mathstrut 1885632q^{23} \) \(\mathstrut -\mathstrut 1449630q^{24} \) \(\mathstrut +\mathstrut 3073221q^{25} \) \(\mathstrut -\mathstrut 8785056q^{26} \) \(\mathstrut +\mathstrut 551880q^{27} \) \(\mathstrut +\mathstrut 3738357q^{28} \) \(\mathstrut +\mathstrut 4081818q^{29} \) \(\mathstrut +\mathstrut 8053200q^{30} \) \(\mathstrut +\mathstrut 2869440q^{31} \) \(\mathstrut +\mathstrut 25221951q^{32} \) \(\mathstrut -\mathstrut 20259792q^{33} \) \(\mathstrut -\mathstrut 3981642q^{34} \) \(\mathstrut +\mathstrut 3731154q^{35} \) \(\mathstrut -\mathstrut 35396379q^{36} \) \(\mathstrut +\mathstrut 1395618q^{37} \) \(\mathstrut +\mathstrut 43479870q^{38} \) \(\mathstrut -\mathstrut 8990688q^{39} \) \(\mathstrut -\mathstrut 82859280q^{40} \) \(\mathstrut -\mathstrut 14420658q^{41} \) \(\mathstrut +\mathstrut 11798514q^{42} \) \(\mathstrut -\mathstrut 61631172q^{43} \) \(\mathstrut +\mathstrut 97011984q^{44} \) \(\mathstrut +\mathstrut 29774682q^{45} \) \(\mathstrut +\mathstrut 89747664q^{46} \) \(\mathstrut -\mathstrut 10368960q^{47} \) \(\mathstrut +\mathstrut 16798782q^{48} \) \(\mathstrut +\mathstrut 17294403q^{49} \) \(\mathstrut +\mathstrut 73325055q^{50} \) \(\mathstrut -\mathstrut 26146728q^{51} \) \(\mathstrut -\mathstrut 80908044q^{52} \) \(\mathstrut +\mathstrut 67502610q^{53} \) \(\mathstrut -\mathstrut 117879300q^{54} \) \(\mathstrut -\mathstrut 105823032q^{55} \) \(\mathstrut +\mathstrut 33746055q^{56} \) \(\mathstrut +\mathstrut 8471112q^{57} \) \(\mathstrut -\mathstrut 159163830q^{58} \) \(\mathstrut -\mathstrut 42590100q^{59} \) \(\mathstrut -\mathstrut 179551008q^{60} \) \(\mathstrut +\mathstrut 191746842q^{61} \) \(\mathstrut -\mathstrut 46983468q^{62} \) \(\mathstrut -\mathstrut 62428401q^{63} \) \(\mathstrut +\mathstrut 7852161q^{64} \) \(\mathstrut +\mathstrut 364283220q^{65} \) \(\mathstrut -\mathstrut 8057952q^{66} \) \(\mathstrut -\mathstrut 255175788q^{67} \) \(\mathstrut +\mathstrut 743485806q^{68} \) \(\mathstrut +\mathstrut 257903856q^{69} \) \(\mathstrut -\mathstrut 234961860q^{70} \) \(\mathstrut +\mathstrut 296514504q^{71} \) \(\mathstrut -\mathstrut 609314265q^{72} \) \(\mathstrut +\mathstrut 344213310q^{73} \) \(\mathstrut -\mathstrut 690696462q^{74} \) \(\mathstrut +\mathstrut 279031116q^{75} \) \(\mathstrut +\mathstrut 728839986q^{76} \) \(\mathstrut -\mathstrut 8269044q^{77} \) \(\mathstrut +\mathstrut 280132776q^{78} \) \(\mathstrut -\mathstrut 960412656q^{79} \) \(\mathstrut -\mathstrut 1333333344q^{80} \) \(\mathstrut -\mathstrut 35827677q^{81} \) \(\mathstrut +\mathstrut 562675302q^{82} \) \(\mathstrut -\mathstrut 1100517180q^{83} \) \(\mathstrut -\mathstrut 255432786q^{84} \) \(\mathstrut +\mathstrut 438179412q^{85} \) \(\mathstrut -\mathstrut 880982256q^{86} \) \(\mathstrut -\mathstrut 621821592q^{87} \) \(\mathstrut +\mathstrut 1206124800q^{88} \) \(\mathstrut +\mathstrut 506816478q^{89} \) \(\mathstrut +\mathstrut 2303452620q^{90} \) \(\mathstrut -\mathstrut 47496582q^{91} \) \(\mathstrut +\mathstrut 691123488q^{92} \) \(\mathstrut +\mathstrut 1693258512q^{93} \) \(\mathstrut +\mathstrut 1388004828q^{94} \) \(\mathstrut -\mathstrut 2203071072q^{95} \) \(\mathstrut +\mathstrut 333385794q^{96} \) \(\mathstrut -\mathstrut 647498250q^{97} \) \(\mathstrut +\mathstrut 121060821q^{98} \) \(\mathstrut -\mathstrut 1900979172q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(426\) \(x\mathstrut +\mathstrut \) \(2016\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{2} + 25 \nu + 276 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + 11 \nu - 288 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(25\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(1706\)\()/6\)

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
18.2745
−22.2358
4.96128
−34.1627 −79.6469 655.088 1423.70 2720.95 2401.00 −4888.28 −13339.4 −48637.4
1.2 13.3607 163.415 −333.491 1922.19 2183.34 2401.00 −11296.4 7021.32 25681.8
1.3 41.8019 0.232339 1235.40 −1791.89 9.71222 2401.00 30239.6 −19682.9 −74904.4
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut -\mathstrut 21 T_{2}^{2} \) \(\mathstrut -\mathstrut 1326 T_{2} \) \(\mathstrut +\mathstrut 19080 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\).