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: \( x^{3} - x^{2} - 426x + 2016 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 7) q^{2} + ( - \beta_{2} - \beta_1 + 28) q^{3} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + (43 \beta_{2} - 13 \beta_1 + 518) q^{5} + (36 \beta_{2} - 6 \beta_1 + 1638) q^{6} + 2401 q^{7} + ( - 470 \beta_{2} + 147 \beta_1 + 4685) q^{8} + (90 \beta_{2} - 126 \beta_1 - 8667) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 7) q^{2} + ( - \beta_{2} - \beta_1 + 28) q^{3} + ( - 8 \beta_{2} + 7 \beta_1 + 519) q^{4} + (43 \beta_{2} - 13 \beta_1 + 518) q^{5} + (36 \beta_{2} - 6 \beta_1 + 1638) q^{6} + 2401 q^{7} + ( - 470 \beta_{2} + 147 \beta_1 + 4685) q^{8} + (90 \beta_{2} - 126 \beta_1 - 8667) q^{9} + (370 \beta_{2} - 470 \beta_1 - 32620) q^{10} + (650 \beta_{2} + 658 \beta_1 - 1148) q^{11} + ( - 700 \beta_{2} + 182 \beta_1 - 35462) q^{12} + (3017 \beta_{2} - 175 \beta_1 - 6594) q^{13} + ( - 2401 \beta_{2} + 16807) q^{14} + ( - 1392 \beta_{2} - 1848 \beta_1 + 66768) q^{15} + ( - 10614 \beta_{2} + 1617 \beta_1 + 160987) q^{16} + (3030 \beta_{2} + 1574 \beta_1 + 338898) q^{17} + (16947 \beta_{2} - 2268 \beta_1 - 91089) q^{18} + ( - 15371 \beta_{2} + 2437 \beta_1 + 74284) q^{19} + (41524 \beta_{2} - 2044 \beta_1 - 640696) q^{20} + ( - 2401 \beta_{2} - 2401 \beta_1 + 67228) q^{21} + ( - 40972 \beta_{2} + 4004 \beta_1 - 949016) q^{22} + ( - 24200 \beta_{2} - 3808 \beta_1 + 628544) q^{23} + (4500 \beta_{2} + 10338 \beta_1 - 483210) q^{24} + ( - 15338 \beta_{2} - 4802 \beta_1 + 1024407) q^{25} + (20986 \beta_{2} - 23394 \beta_1 - 2928352) q^{26} + (33930 \beta_{2} + 15786 \beta_1 + 183960) q^{27} + ( - 19208 \beta_{2} + 16807 \beta_1 + 1246119) q^{28} + (54866 \beta_{2} + 18914 \beta_1 + 1360606) q^{29} + (51960 \beta_{2} - 14280 \beta_1 + 2684400) q^{30} + (55698 \beta_{2} - 70302 \beta_1 + 956480) q^{31} + ( - 36066 \beta_{2} + 20055 \beta_1 + 8407317) q^{32} + ( - 76152 \beta_{2} + 65640 \beta_1 - 6753264) q^{33} + ( - 438178 \beta_{2} - 748 \beta_1 - 1327214) q^{34} + (103243 \beta_{2} - 31213 \beta_1 + 1243718) q^{35} + (209376 \beta_{2} - 83601 \beta_1 - 11798793) q^{36} + (209418 \beta_{2} + 60522 \beta_1 + 465206) q^{37} + ( - 248060 \beta_{2} + 139278 \beta_1 + 14493290) q^{38} + ( - 110012 \beta_{2} - 13748 \beta_1 - 2996896) q^{39} + (625640 \beta_{2} - 76600 \beta_1 - 27619760) q^{40} + ( - 163478 \beta_{2} - 131894 \beta_1 - 4806886) q^{41} + (86436 \beta_{2} - 14406 \beta_1 + 3932838) q^{42} + (121982 \beta_{2} + 65366 \beta_1 - 20543724) q^{43} + (314984 \beta_{2} + 1960 \beta_1 + 32337328) q^{44} + ( - 714681 \beta_{2} + 7551 \beta_1 + 9924894) q^{45} + ( - 405224 \beta_{2} + 119896 \beta_1 + 29915888) q^{46} + ( - 534778 \beta_{2} + 83238 \beta_1 - 3456320) q^{47} + (174140 \beta_{2} + 9710 \beta_1 + 5599594) q^{48} + 5764801 q^{49} + ( - 727615 \beta_{2} + 44940 \beta_1 + 24441685) q^{50} + ( - 587238 \beta_{2} - 186102 \beta_1 - 8715576) q^{51} + (2925244 \beta_{2} - 361424 \beta_1 - 26969348) q^{52} + (1553376 \beta_{2} - 450352 \beta_1 + 22500870) q^{53} + ( - 1176120 \beta_{2} - 32292 \beta_1 - 39293100) q^{54} + (1659356 \beta_{2} + 988364 \beta_1 - 35274344) q^{55} + ( - 1128470 \beta_{2} + 352947 \beta_1 + 11248685) q^{56} + (403894 \beta_{2} + 182350 \beta_1 + 2823704) q^{57} + ( - 2535150 \beta_{2} - 138180 \beta_1 - 53054610) q^{58} + ( - 2231195 \beta_{2} - 49659 \beta_1 - 14196700) q^{59} + ( - 991536 \beta_{2} + 396816 \beta_1 - 59850336) q^{60} + (589107 \beta_{2} - 844773 \beta_1 + 63915614) q^{61} + (3668848 \beta_{2} - 1303812 \beta_1 - 15661156) q^{62} + (216090 \beta_{2} - 302526 \beta_1 - 20809467) q^{63} + ( - 4312590 \beta_{2} - 314727 \beta_1 + 2617387) q^{64} + ( - 958090 \beta_{2} + 1035790 \beta_1 + 121427740) q^{65} + (2410512 \beta_{2} + 1386384 \beta_1 - 2685984) q^{66} + (3939816 \beta_{2} + 47712 \beta_1 - 85058596) q^{67} + ( - 613704 \beta_{2} + 2251634 \beta_1 + 247828602) q^{68} + (697656 \beta_{2} - 981336 \beta_1 + 85967952) q^{69} + (888370 \beta_{2} - 1128470 \beta_1 - 78320620) q^{70} + (3499356 \beta_{2} - 526260 \beta_1 + 98838168) q^{71} + (8765370 \beta_{2} - 1391229 \beta_1 - 203104755) q^{72} + ( - 9544844 \beta_{2} + 118516 \beta_1 + 114737770) q^{73} + ( - 4189718 \beta_{2} - 679140 \beta_1 - 230232154) q^{74} + ( - 4723 \beta_{2} - 1484467 \beta_1 + 93010372) q^{75} + ( - 15924468 \beta_{2} + 2299290 \beta_1 + 242946662) q^{76} + (1560650 \beta_{2} + 1579858 \beta_1 - 2756348) q^{77} + (3780504 \beta_{2} + 591360 \beta_1 + 93377592) q^{78} + ( - 7475532 \beta_{2} + 1679412 \beta_1 - 320137552) q^{79} + (11964112 \beta_{2} - 4328752 \beta_1 - 444444448) q^{80} + ( - 4598370 \beta_{2} + 3824982 \beta_1 - 11942559) q^{81} + (13216518 \beta_{2} - 570276 \beta_1 + 187558434) q^{82} + ( - 4479559 \beta_{2} - 1977367 \beta_1 - 366839060) q^{83} + ( - 1680700 \beta_{2} + 436982 \beta_1 - 85144262) q^{84} + (18558254 \beta_{2} - 1518034 \beta_1 + 146059804) q^{85} + (16416916 \beta_{2} - 4116 \beta_1 - 293660752) q^{86} + ( - 5138394 \beta_{2} + 457014 \beta_1 - 207273864) q^{87} + ( - 11172080 \beta_{2} - 4229456 \beta_1 + 402041600) q^{88} + ( - 10612104 \beta_{2} + 1815976 \beta_1 + 168938826) q^{89} + ( - 11130390 \beta_{2} + 5100930 \beta_1 + 767817540) q^{90} + (7243817 \beta_{2} - 420175 \beta_1 - 15832194) q^{91} + ( - 25723952 \beta_{2} + 6344912 \beta_1 + 230374496) q^{92} + (1921156 \beta_{2} - 7972076 \beta_1 + 564419504) q^{93} + ( - 2488928 \beta_{2} + 4825540 \beta_1 + 462668276) q^{94} + (10749416 \beta_{2} - 4098416 \beta_1 - 734357024) q^{95} + ( - 8360604 \beta_{2} - 6385806 \beta_1 + 111128598) q^{96} + (33276782 \beta_{2} - 864850 \beta_1 - 215832750) q^{97} + ( - 5764801 \beta_{2} + 40353607) q^{98} + ( - 7689150 \beta_{2} + 376362 \beta_1 - 633659724) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 21 q^{2} + 84 q^{3} + 1557 q^{4} + 1554 q^{5} + 4914 q^{6} + 7203 q^{7} + 14055 q^{8} - 26001 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 21 q^{2} + 84 q^{3} + 1557 q^{4} + 1554 q^{5} + 4914 q^{6} + 7203 q^{7} + 14055 q^{8} - 26001 q^{9} - 97860 q^{10} - 3444 q^{11} - 106386 q^{12} - 19782 q^{13} + 50421 q^{14} + 200304 q^{15} + 482961 q^{16} + 1016694 q^{17} - 273267 q^{18} + 222852 q^{19} - 1922088 q^{20} + 201684 q^{21} - 2847048 q^{22} + 1885632 q^{23} - 1449630 q^{24} + 3073221 q^{25} - 8785056 q^{26} + 551880 q^{27} + 3738357 q^{28} + 4081818 q^{29} + 8053200 q^{30} + 2869440 q^{31} + 25221951 q^{32} - 20259792 q^{33} - 3981642 q^{34} + 3731154 q^{35} - 35396379 q^{36} + 1395618 q^{37} + 43479870 q^{38} - 8990688 q^{39} - 82859280 q^{40} - 14420658 q^{41} + 11798514 q^{42} - 61631172 q^{43} + 97011984 q^{44} + 29774682 q^{45} + 89747664 q^{46} - 10368960 q^{47} + 16798782 q^{48} + 17294403 q^{49} + 73325055 q^{50} - 26146728 q^{51} - 80908044 q^{52} + 67502610 q^{53} - 117879300 q^{54} - 105823032 q^{55} + 33746055 q^{56} + 8471112 q^{57} - 159163830 q^{58} - 42590100 q^{59} - 179551008 q^{60} + 191746842 q^{61} - 46983468 q^{62} - 62428401 q^{63} + 7852161 q^{64} + 364283220 q^{65} - 8057952 q^{66} - 255175788 q^{67} + 743485806 q^{68} + 257903856 q^{69} - 234961860 q^{70} + 296514504 q^{71} - 609314265 q^{72} + 344213310 q^{73} - 690696462 q^{74} + 279031116 q^{75} + 728839986 q^{76} - 8269044 q^{77} + 280132776 q^{78} - 960412656 q^{79} - 1333333344 q^{80} - 35827677 q^{81} + 562675302 q^{82} - 1100517180 q^{83} - 255432786 q^{84} + 438179412 q^{85} - 880982256 q^{86} - 621821592 q^{87} + 1206124800 q^{88} + 506816478 q^{89} + 2303452620 q^{90} - 47496582 q^{91} + 691123488 q^{92} + 1693258512 q^{93} + 1388004828 q^{94} - 2203071072 q^{95} + 333385794 q^{96} - 647498250 q^{97} + 121060821 q^{98} - 1900979172 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 25\nu + 276 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 11\nu - 288 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 25\beta_{2} - 11\beta _1 + 1706 ) / 6 \) Copy content Toggle raw display

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 \( T_{2}^{3} - 21T_{2}^{2} - 1326T_{2} + 19080 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 21 T^{2} - 1326 T + 19080 \) Copy content Toggle raw display
$3$ \( T^{3} - 84 T^{2} - 12996 T + 3024 \) Copy content Toggle raw display
$5$ \( T^{3} - 1554 T^{2} + \cdots + 4903718400 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots + 108859759460352 \) Copy content Toggle raw display
$13$ \( T^{3} + 19782 T^{2} + \cdots - 41548412541440 \) Copy content Toggle raw display
$17$ \( T^{3} - 1016694 T^{2} + \cdots - 21\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{3} - 222852 T^{2} + \cdots - 43\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{3} - 1885632 T^{2} + \cdots + 97\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{3} - 4081818 T^{2} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} - 2869440 T^{2} + \cdots - 74\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{3} - 1395618 T^{2} + \cdots - 34\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{3} + 14420658 T^{2} + \cdots - 19\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{3} + 61631172 T^{2} + \cdots + 68\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{3} + 10368960 T^{2} + \cdots - 43\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{3} - 67502610 T^{2} + \cdots + 23\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{3} + 42590100 T^{2} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} - 191746842 T^{2} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{3} + 255175788 T^{2} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{3} - 296514504 T^{2} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{3} - 344213310 T^{2} + \cdots + 19\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{3} + 960412656 T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + 1100517180 T^{2} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{3} - 506816478 T^{2} + \cdots + 19\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{3} + 647498250 T^{2} + \cdots - 49\!\cdots\!16 \) Copy content Toggle raw display
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