Properties

Label 7.8.a.b
Level $7$
Weight $8$
Character orbit 7.a
Self dual yes
Analytic conductor $2.187$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.18669517839\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{865}) \)
Defining polynomial: \( x^{2} - x - 216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{865})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta + 46) q^{3} + (3 \beta + 89) q^{4} + (10 \beta + 160) q^{5} + ( - 50 \beta - 478) q^{6} - 343 q^{7} + (33 \beta - 609) q^{8} + (188 \beta + 793) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (2 \beta + 46) q^{3} + (3 \beta + 89) q^{4} + (10 \beta + 160) q^{5} + ( - 50 \beta - 478) q^{6} - 343 q^{7} + (33 \beta - 609) q^{8} + (188 \beta + 793) q^{9} + ( - 180 \beta - 2320) q^{10} + ( - 116 \beta + 1480) q^{11} + (322 \beta + 5390) q^{12} + ( - 882 \beta + 1708) q^{13} + (343 \beta + 343) q^{14} + (800 \beta + 11680) q^{15} + (159 \beta - 17911) q^{16} + (324 \beta - 906) q^{17} + ( - 1169 \beta - 41401) q^{18} + ( - 858 \beta + 16834) q^{19} + (1400 \beta + 20720) q^{20} + ( - 686 \beta - 15778) q^{21} + ( - 1248 \beta + 23576) q^{22} + (48 \beta - 3312) q^{23} + (366 \beta - 13758) q^{24} + (3300 \beta - 30925) q^{25} + (56 \beta + 188804) q^{26} + (6236 \beta + 17092) q^{27} + ( - 1029 \beta - 30527) q^{28} + ( - 9380 \beta + 15010) q^{29} + ( - 13280 \beta - 184480) q^{30} + (2508 \beta - 197172) q^{31} + (13369 \beta + 61519) q^{32} + ( - 2608 \beta + 17968) q^{33} + (258 \beta - 69078) q^{34} + ( - 3430 \beta - 54880) q^{35} + (19675 \beta + 192401) q^{36} + (27180 \beta + 170106) q^{37} + ( - 15118 \beta + 168494) q^{38} + ( - 38920 \beta - 302456) q^{39} + ( - 480 \beta - 26160) q^{40} + ( - 23716 \beta + 379190) q^{41} + (17150 \beta + 163954) q^{42} + ( - 5628 \beta - 237424) q^{43} + ( - 6232 \beta + 56552) q^{44} + (39890 \beta + 532960) q^{45} + (3216 \beta - 7056) q^{46} + (36900 \beta - 563004) q^{47} + ( - 28190 \beta - 755218) q^{48} + 117649 q^{49} + (24325 \beta - 681875) q^{50} + (13740 \beta + 98292) q^{51} + ( - 76020 \beta - 419524) q^{52} + ( - 5184 \beta + 1432014) q^{53} + ( - 29564 \beta - 1364068) q^{54} + ( - 4920 \beta - 13760) q^{55} + ( - 11319 \beta + 208887) q^{56} + ( - 7516 \beta + 403708) q^{57} + (3750 \beta + 2011070) q^{58} + (53622 \beta + 53274) q^{59} + (108640 \beta + 1557920) q^{60} + ( - 57558 \beta - 403544) q^{61} + (192156 \beta - 344556) q^{62} + ( - 64484 \beta - 271999) q^{63} + ( - 108609 \beta - 656615) q^{64} + ( - 132860 \beta - 1631840) q^{65} + ( - 12752 \beta + 545360) q^{66} + (50928 \beta - 189788) q^{67} + (27090 \beta + 129318) q^{68} + ( - 4320 \beta - 131616) q^{69} + (61740 \beta + 795760) q^{70} + ( - 130872 \beta - 3684672) q^{71} + ( - 82119 \beta + 857127) q^{72} + (191928 \beta + 2054658) q^{73} + ( - 224466 \beta - 6040986) q^{74} + (96550 \beta + 3050) q^{75} + ( - 28434 \beta + 942242) q^{76} + (39788 \beta - 507640) q^{77} + (380296 \beta + 8709176) q^{78} + (277080 \beta - 3342760) q^{79} + ( - 152080 \beta - 2522320) q^{80} + ( - 77644 \beta + 1745893) q^{81} + ( - 331758 \beta + 4743466) q^{82} + ( - 132594 \beta + 5895834) q^{83} + ( - 110446 \beta - 1848770) q^{84} + (46020 \beta + 554880) q^{85} + (248680 \beta + 1453072) q^{86} + ( - 420220 \beta - 3361700) q^{87} + (115656 \beta - 1728168) q^{88} + (363184 \beta + 4704538) q^{89} + ( - 612740 \beta - 9149200) q^{90} + (302526 \beta - 585844) q^{91} + ( - 5520 \beta - 263664) q^{92} + ( - 273960 \beta - 7986456) q^{93} + (489204 \beta - 7407396) q^{94} + (22480 \beta + 840160) q^{95} + (764750 \beta + 8605282) q^{96} + ( - 94668 \beta + 5428710) q^{97} + ( - 117649 \beta - 117649) q^{98} + (164444 \beta - 3536888) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 94 q^{3} + 181 q^{4} + 330 q^{5} - 1006 q^{6} - 686 q^{7} - 1185 q^{8} + 1774 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 94 q^{3} + 181 q^{4} + 330 q^{5} - 1006 q^{6} - 686 q^{7} - 1185 q^{8} + 1774 q^{9} - 4820 q^{10} + 2844 q^{11} + 11102 q^{12} + 2534 q^{13} + 1029 q^{14} + 24160 q^{15} - 35663 q^{16} - 1488 q^{17} - 83971 q^{18} + 32810 q^{19} + 42840 q^{20} - 32242 q^{21} + 45904 q^{22} - 6576 q^{23} - 27150 q^{24} - 58550 q^{25} + 377664 q^{26} + 40420 q^{27} - 62083 q^{28} + 20640 q^{29} - 382240 q^{30} - 391836 q^{31} + 136407 q^{32} + 33328 q^{33} - 137898 q^{34} - 113190 q^{35} + 404477 q^{36} + 367392 q^{37} + 321870 q^{38} - 643832 q^{39} - 52800 q^{40} + 734664 q^{41} + 345058 q^{42} - 480476 q^{43} + 106872 q^{44} + 1105810 q^{45} - 10896 q^{46} - 1089108 q^{47} - 1538626 q^{48} + 235298 q^{49} - 1339425 q^{50} + 210324 q^{51} - 915068 q^{52} + 2858844 q^{53} - 2757700 q^{54} - 32440 q^{55} + 406455 q^{56} + 799900 q^{57} + 4025890 q^{58} + 160170 q^{59} + 3224480 q^{60} - 864646 q^{61} - 496956 q^{62} - 608482 q^{63} - 1421839 q^{64} - 3396540 q^{65} + 1077968 q^{66} - 328648 q^{67} + 285726 q^{68} - 267552 q^{69} + 1653260 q^{70} - 7500216 q^{71} + 1632135 q^{72} + 4301244 q^{73} - 12306438 q^{74} + 102650 q^{75} + 1856050 q^{76} - 975492 q^{77} + 17798648 q^{78} - 6408440 q^{79} - 5196720 q^{80} + 3414142 q^{81} + 9155174 q^{82} + 11659074 q^{83} - 3807986 q^{84} + 1155780 q^{85} + 3154824 q^{86} - 7143620 q^{87} - 3340680 q^{88} + 9772260 q^{89} - 18911140 q^{90} - 869162 q^{91} - 532848 q^{92} - 16246872 q^{93} - 14325588 q^{94} + 1702800 q^{95} + 17975314 q^{96} + 10762752 q^{97} - 352947 q^{98} - 6909332 q^{99}+O(q^{100}) \) 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
15.2054
−14.2054
−16.2054 76.4109 134.616 312.054 −1238.27 −343.000 −107.220 3651.62 −5056.98
1.2 13.2054 17.5891 46.3837 17.9456 232.272 −343.000 −1077.78 −1877.62 236.979
\(n\): 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 7.8.a.b 2
3.b odd 2 1 63.8.a.e 2
4.b odd 2 1 112.8.a.f 2
5.b even 2 1 175.8.a.c 2
5.c odd 4 2 175.8.b.b 4
7.b odd 2 1 49.8.a.c 2
7.c even 3 2 49.8.c.e 4
7.d odd 6 2 49.8.c.f 4
8.b even 2 1 448.8.a.k 2
8.d odd 2 1 448.8.a.t 2
21.c even 2 1 441.8.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.8.a.b 2 1.a even 1 1 trivial
49.8.a.c 2 7.b odd 2 1
49.8.c.e 4 7.c even 3 2
49.8.c.f 4 7.d odd 6 2
63.8.a.e 2 3.b odd 2 1
112.8.a.f 2 4.b odd 2 1
175.8.a.c 2 5.b even 2 1
175.8.b.b 4 5.c odd 4 2
441.8.a.l 2 21.c even 2 1
448.8.a.k 2 8.b even 2 1
448.8.a.t 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} - 214 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 214 \) Copy content Toggle raw display
$3$ \( T^{2} - 94T + 1344 \) Copy content Toggle raw display
$5$ \( T^{2} - 330T + 5600 \) Copy content Toggle raw display
$7$ \( (T + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2844 T - 887776 \) Copy content Toggle raw display
$13$ \( T^{2} - 2534 T - 166620776 \) Copy content Toggle raw display
$17$ \( T^{2} + 1488 T - 22147524 \) Copy content Toggle raw display
$19$ \( T^{2} - 32810 T + 109928560 \) Copy content Toggle raw display
$23$ \( T^{2} + 6576 T + 10312704 \) Copy content Toggle raw display
$29$ \( T^{2} - 20640 T - 18920124100 \) Copy content Toggle raw display
$31$ \( T^{2} + 391836 T + 37023636384 \) Copy content Toggle raw display
$37$ \( T^{2} - 367392 T - 126010986084 \) Copy content Toggle raw display
$41$ \( T^{2} - 734664 T + 13303276364 \) Copy content Toggle raw display
$43$ \( T^{2} + 480476 T + 50864711104 \) Copy content Toggle raw display
$47$ \( T^{2} + 1089108 T + 2090896416 \) Copy content Toggle raw display
$53$ \( T^{2} - 2858844 T + 2037435782724 \) Copy content Toggle raw display
$59$ \( T^{2} - 160170 T - 615374101440 \) Copy content Toggle raw display
$61$ \( T^{2} + 864646 T - 529516501136 \) Copy content Toggle raw display
$67$ \( T^{2} + 328648 T - 533876854064 \) Copy content Toggle raw display
$71$ \( T^{2} + 7500216 T + 10359492378624 \) Copy content Toggle raw display
$73$ \( T^{2} - 4301244 T - 3340687254156 \) Copy content Toggle raw display
$79$ \( T^{2} + 6408440 T - 6335206025600 \) Copy content Toggle raw display
$83$ \( T^{2} - 11659074 T + 30181573873584 \) Copy content Toggle raw display
$89$ \( T^{2} - 9772260 T - 4649674734460 \) Copy content Toggle raw display
$97$ \( T^{2} - 10762752 T + 27021168617436 \) Copy content Toggle raw display
show more
show less