# 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

## 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: $$x^{2} - x - 216$$ x^2 - x - 216 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{8}^{\mathrm{new}}(\Gamma_0(7))$$.

## Hecke characteristic polynomials

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