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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 426 x + 2016\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 21 q^{2} + 84 q^{3} + 1557 q^{4} + 1554 q^{5} + 4914 q^{6} + 7203 q^{7} + 14055 q^{8} - 26001 q^{9} + O(q^{10}) \) \( 3 q + 21 q^{2} + 84 q^{3} + 1557 q^{4} + 1554 q^{5} + 4914 q^{6} + 7203 q^{7} + 14055 q^{8} - 26001 q^{9} - 97860 q^{10} - 3444 q^{11} - 106386 q^{12} - 19782 q^{13} + 50421 q^{14} + 200304 q^{15} + 482961 q^{16} + 1016694 q^{17} - 273267 q^{18} + 222852 q^{19} - 1922088 q^{20} + 201684 q^{21} - 2847048 q^{22} + 1885632 q^{23} - 1449630 q^{24} + 3073221 q^{25} - 8785056 q^{26} + 551880 q^{27} + 3738357 q^{28} + 4081818 q^{29} + 8053200 q^{30} + 2869440 q^{31} + 25221951 q^{32} - 20259792 q^{33} - 3981642 q^{34} + 3731154 q^{35} - 35396379 q^{36} + 1395618 q^{37} + 43479870 q^{38} - 8990688 q^{39} - 82859280 q^{40} - 14420658 q^{41} + 11798514 q^{42} - 61631172 q^{43} + 97011984 q^{44} + 29774682 q^{45} + 89747664 q^{46} - 10368960 q^{47} + 16798782 q^{48} + 17294403 q^{49} + 73325055 q^{50} - 26146728 q^{51} - 80908044 q^{52} + 67502610 q^{53} - 117879300 q^{54} - 105823032 q^{55} + 33746055 q^{56} + 8471112 q^{57} - 159163830 q^{58} - 42590100 q^{59} - 179551008 q^{60} + 191746842 q^{61} - 46983468 q^{62} - 62428401 q^{63} + 7852161 q^{64} + 364283220 q^{65} - 8057952 q^{66} - 255175788 q^{67} + 743485806 q^{68} + 257903856 q^{69} - 234961860 q^{70} + 296514504 q^{71} - 609314265 q^{72} + 344213310 q^{73} - 690696462 q^{74} + 279031116 q^{75} + 728839986 q^{76} - 8269044 q^{77} + 280132776 q^{78} - 960412656 q^{79} - 1333333344 q^{80} - 35827677 q^{81} + 562675302 q^{82} - 1100517180 q^{83} - 255432786 q^{84} + 438179412 q^{85} - 880982256 q^{86} - 621821592 q^{87} + 1206124800 q^{88} + 506816478 q^{89} + 2303452620 q^{90} - 47496582 q^{91} + 691123488 q^{92} + 1693258512 q^{93} + 1388004828 q^{94} - 2203071072 q^{95} + 333385794 q^{96} - 647498250 q^{97} + 121060821 q^{98} - 1900979172 q^{99} + O(q^{100}) \)

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

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

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:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 21 T_{2}^{2} - 1326 T_{2} + 19080 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 19080 - 1326 T - 21 T^{2} + T^{3} \)
$3$ \( 3024 - 12996 T - 84 T^{2} + T^{3} \)
$5$ \( 4903718400 - 3258840 T - 1554 T^{2} + T^{3} \)
$7$ \( ( -2401 + T )^{3} \)
$11$ \( 108859759460352 - 6618499968 T + 3444 T^{2} + T^{3} \)
$13$ \( -41548412541440 - 12931283064 T + 19782 T^{2} + T^{3} \)
$17$ \( -21973894921381032 + 293494511292 T - 1016694 T^{2} + T^{3} \)
$19$ \( -43011870587515760 - 353981719620 T - 222852 T^{2} + T^{3} \)
$23$ \( 974648214470209536 + 14194696128 T - 1885632 T^{2} + T^{3} \)
$29$ \( 4423213168251517800 - 4782422143620 T - 4081818 T^{2} + T^{3} \)
$31$ \( -74172820551747190784 - 58176366315792 T - 2869440 T^{2} + T^{3} \)
$37$ \( -\)\(34\!\cdots\!28\)\( - 127209247191204 T - 1395618 T^{2} + T^{3} \)
$41$ \( -\)\(19\!\cdots\!12\)\( - 217166148381924 T + 14420658 T^{2} + T^{3} \)
$43$ \( \)\(68\!\cdots\!80\)\( + 1179825167354496 T + 61631172 T^{2} + T^{3} \)
$47$ \( -\)\(43\!\cdots\!16\)\( - 410564457968592 T + 10368960 T^{2} + T^{3} \)
$53$ \( \)\(23\!\cdots\!28\)\( - 3604244065118868 T - 67502610 T^{2} + T^{3} \)
$59$ \( \)\(42\!\cdots\!00\)\( - 6912010951598820 T + 42590100 T^{2} + T^{3} \)
$61$ \( \)\(51\!\cdots\!08\)\( + 3514240390404936 T - 191746842 T^{2} + T^{3} \)
$67$ \( -\)\(20\!\cdots\!64\)\( - 1447084483230480 T + 255175788 T^{2} + T^{3} \)
$71$ \( \)\(16\!\cdots\!80\)\( + 10350223039033344 T - 296514504 T^{2} + T^{3} \)
$73$ \( \)\(19\!\cdots\!48\)\( - 93316419822721428 T - 344213310 T^{2} + T^{3} \)
$79$ \( -\)\(11\!\cdots\!00\)\( + 207231646446206400 T + 960412656 T^{2} + T^{3} \)
$83$ \( \)\(18\!\cdots\!48\)\( + 313075935079720092 T + 1100517180 T^{2} + T^{3} \)
$89$ \( \)\(19\!\cdots\!40\)\( - 94874685327766740 T - 506816478 T^{2} + T^{3} \)
$97$ \( -\)\(49\!\cdots\!16\)\( - 1460996931372270852 T + 647498250 T^{2} + T^{3} \)
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