# Properties

 Label 464.6.a.k Level $464$ Weight $6$ Character orbit 464.a Self dual yes Analytic conductor $74.418$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$464 = 2^{4} \cdot 29$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 464.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$74.4180923932$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902$$ x^7 - 3*x^6 - 184*x^5 + 584*x^4 + 10145*x^3 - 34491*x^2 - 149754*x + 524902 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - 4) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} + 5) q^{5} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 26) q^{7} + ( - \beta_{6} + 3 \beta_{5} - 5 \beta_{4} - \beta_{2} + 2 \beta_1 + 146) q^{9}+O(q^{10})$$ q + (b4 - 4) * q^3 + (b5 + b3 - b2 + 5) * q^5 + (-b6 - 2*b4 + b2 - 26) * q^7 + (-b6 + 3*b5 - 5*b4 - b2 + 2*b1 + 146) * q^9 $$q + (\beta_{4} - 4) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} + 5) q^{5} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 26) q^{7} + ( - \beta_{6} + 3 \beta_{5} - 5 \beta_{4} - \beta_{2} + 2 \beta_1 + 146) q^{9} + (\beta_{6} + 3 \beta_{5} + 9 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 160) q^{11} + (\beta_{6} - 8 \beta_{5} - 10 \beta_{4} - 5 \beta_{3} + \beta_{2} + 3 \beta_1 + 58) q^{13} + (6 \beta_{6} - 11 \beta_{5} + 17 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} - 12 \beta_1 + 82) q^{15} + (3 \beta_{6} - 20 \beta_{5} + 5 \beta_{4} - 10 \beta_{3} - \beta_{2} - 13 \beta_1 - 131) q^{17} + (4 \beta_{6} - 12 \beta_{5} - 35 \beta_{4} + 10 \beta_{3} - 16 \beta_{2} + \cdots - 607) q^{19}+ \cdots + (299 \beta_{6} - 2039 \beta_{5} + 5006 \beta_{4} - 399 \beta_{3} + \cdots - 45282) q^{99}+O(q^{100})$$ q + (b4 - 4) * q^3 + (b5 + b3 - b2 + 5) * q^5 + (-b6 - 2*b4 + b2 - 26) * q^7 + (-b6 + 3*b5 - 5*b4 - b2 + 2*b1 + 146) * q^9 + (b6 + 3*b5 + 9*b4 + 3*b3 + 3*b2 - 160) * q^11 + (b6 - 8*b5 - 10*b4 - 5*b3 + b2 + 3*b1 + 58) * q^13 + (6*b6 - 11*b5 + 17*b4 - 11*b3 + 2*b2 - 12*b1 + 82) * q^15 + (3*b6 - 20*b5 + 5*b4 - 10*b3 - b2 - 13*b1 - 131) * q^17 + (4*b6 - 12*b5 - 35*b4 + 10*b3 - 16*b2 + 5*b1 - 607) * q^19 + (14*b6 - 21*b5 + 10*b4 - 15*b3 + 22*b2 + 8*b1 - 612) * q^21 + (-8*b6 + 3*b5 - 33*b4 - 4*b3 - b2 - 2*b1 + 658) * q^23 + (5*b6 + 5*b5 - 26*b3 - 15*b2 - 5*b1 + 799) * q^25 + (9*b6 - 109*b5 + 61*b4 - 33*b3 + 99*b2 - 8*b1 - 920) * q^27 + 841 * q^29 + (9*b6 + 78*b5 - 217*b4 - 7*b3 - 2*b2 - 17*b1 - 1013) * q^31 + (-17*b6 + 93*b5 - 202*b4 - 39*b3 + 12*b2 - 2*b1 + 5015) * q^33 + (27*b6 + 78*b5 - 294*b4 - 79*b3 - 28*b2 + 49*b1 - 883) * q^35 + (-23*b6 - 87*b5 + 97*b4 + 100*b3 + 50*b2 + 62*b1 + 628) * q^37 + (-59*b6 - 56*b5 + 55*b4 + 43*b3 - 14*b2 + 43*b1 - 4829) * q^39 + (-24*b6 - 153*b5 - 116*b4 + 217*b2 + 3*b1 + 2765) * q^41 + (-60*b6 + 52*b5 + 24*b4 - 30*b3 + 148*b2 + 25*b1 - 2799) * q^43 + (-41*b6 + 259*b5 - 676*b4 + 121*b3 - 183*b2 + 22*b1 + 1426) * q^45 + (-37*b6 - 122*b5 - 128*b4 - 54*b3 - 85*b2 + 85*b1 - 2053) * q^47 + (22*b6 - 198*b5 - 322*b4 - 215*b3 + 21*b2 - 97*b1 + 5542) * q^49 + (12*b6 + 403*b5 - 903*b4 + 335*b3 - 538*b2 + 37*b1 - 2957) * q^51 + (81*b6 + 366*b5 - 634*b4 + 150*b3 - 28*b2 - 18*b1 - 8043) * q^53 + (100*b6 - 173*b5 + 28*b4 - 228*b3 + 443*b2 - 218*b1 + 3586) * q^55 + (-38*b6 - 228*b5 - 689*b4 - 104*b3 - 238*b2 - 133*b1 - 12487) * q^57 + (9*b6 - 554*b5 + 268*b4 - 15*b3 + 110*b2 - 143*b1 - 2147) * q^59 + (-27*b6 + 162*b5 + 645*b4 - 235*b3 + 98*b2 - 182*b1 + 14622) * q^61 + (-151*b6 + 491*b5 - 695*b4 + 216*b3 - 374*b2 + 180*b1 + 13046) * q^63 + (-199*b6 - 295*b5 - 69*b4 + 398*b3 - 41*b2 + 138*b1 - 21521) * q^65 + (69*b6 - 433*b5 + 877*b4 - 150*b3 - 228*b2 - 322*b1 - 15040) * q^67 + (163*b6 - 306*b5 + 1001*b4 - 13*b3 + 176*b2 - 12*b1 - 15736) * q^69 + (100*b6 + 64*b5 - 490*b4 - 354*b3 + 654*b2 + 82*b1 + 7498) * q^71 + (185*b6 + 471*b5 - 331*b4 + 470*b3 + 178*b2 + 34*b1 - 2284) * q^73 + (-37*b6 - 49*b5 + 725*b4 + 529*b3 - 3*b2 - 41*b1 - 5925) * q^75 + (309*b6 - 352*b5 + 1249*b4 - 365*b3 - 530*b2 + 202*b1 - 13808) * q^77 + (62*b6 + 745*b5 - 1531*b4 + 557*b3 - 274*b2 + 260*b1 - 29606) * q^79 + (-354*b6 + 1264*b5 - 2111*b4 + 369*b3 - 921*b2 + 488*b1 - 17223) * q^81 + (404*b6 + 1259*b5 - 2261*b4 + 299*b3 - 502*b2 - 165*b1 + 18857) * q^83 + (-22*b6 - 379*b5 + 3744*b4 + 189*b3 - 334*b2 + 392*b1 - 16890) * q^85 + (841*b4 - 3364) * q^87 + (-66*b6 + 753*b5 - 968*b4 - 996*b3 - 695*b2 + 441*b1 - 8733) * q^89 + (-298*b6 + 999*b5 + 1897*b4 + 1099*b3 - 172*b2 - 145*b1 - 28035) * q^91 + (390*b6 - 303*b5 - 676*b4 + 416*b3 + 856*b2 - 883*b1 - 67690) * q^93 + (-411*b6 - 1931*b5 + 780*b4 - 230*b3 + 452*b2 + 525*b1 + 17595) * q^95 + (63*b6 + 472*b5 - 821*b4 + 886*b3 + 717*b2 - 135*b1 - 16647) * q^97 + (299*b6 - 2039*b5 + 5006*b4 - 399*b3 + 1321*b2 - 446*b1 - 45282) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9}+O(q^{10})$$ 7 * q - 26 * q^3 + 32 * q^5 - 184 * q^7 + 1005 * q^9 $$7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9} - 1106 q^{11} + 408 q^{13} + 614 q^{15} - 874 q^{17} - 4288 q^{19} - 4200 q^{21} + 4532 q^{23} + 5527 q^{25} - 5942 q^{27} + 5887 q^{29} - 7794 q^{31} + 34410 q^{33} - 7088 q^{35} + 5086 q^{37} - 33394 q^{39} + 19826 q^{41} - 19498 q^{43} + 7854 q^{45} - 14278 q^{47} + 38431 q^{49} - 23892 q^{51} - 58644 q^{53} + 25574 q^{55} - 88540 q^{57} - 12888 q^{59} + 102866 q^{61} + 88632 q^{63} - 149206 q^{65} - 102996 q^{67} - 107244 q^{69} + 51596 q^{71} - 17566 q^{73} - 39356 q^{75} - 94104 q^{77} - 212058 q^{79} - 128285 q^{81} + 122928 q^{83} - 109336 q^{85} - 21866 q^{87} - 66510 q^{89} - 194368 q^{91} - 474274 q^{93} + 131676 q^{95} - 118182 q^{97} - 300668 q^{99}+O(q^{100})$$ 7 * q - 26 * q^3 + 32 * q^5 - 184 * q^7 + 1005 * q^9 - 1106 * q^11 + 408 * q^13 + 614 * q^15 - 874 * q^17 - 4288 * q^19 - 4200 * q^21 + 4532 * q^23 + 5527 * q^25 - 5942 * q^27 + 5887 * q^29 - 7794 * q^31 + 34410 * q^33 - 7088 * q^35 + 5086 * q^37 - 33394 * q^39 + 19826 * q^41 - 19498 * q^43 + 7854 * q^45 - 14278 * q^47 + 38431 * q^49 - 23892 * q^51 - 58644 * q^53 + 25574 * q^55 - 88540 * q^57 - 12888 * q^59 + 102866 * q^61 + 88632 * q^63 - 149206 * q^65 - 102996 * q^67 - 107244 * q^69 + 51596 * q^71 - 17566 * q^73 - 39356 * q^75 - 94104 * q^77 - 212058 * q^79 - 128285 * q^81 + 122928 * q^83 - 109336 * q^85 - 21866 * q^87 - 66510 * q^89 - 194368 * q^91 - 474274 * q^93 + 131676 * q^95 - 118182 * q^97 - 300668 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} - 18\nu^{5} - 126\nu^{4} + 2142\nu^{3} + 3339\nu^{2} - 43244\nu + 12946 ) / 960$$ (v^6 - 18*v^5 - 126*v^4 + 2142*v^3 + 3339*v^2 - 43244*v + 12946) / 960 $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 8\nu^{5} - 176\nu^{4} + 952\nu^{3} + 9769\nu^{2} - 24904\nu - 146514 ) / 960$$ (v^6 - 8*v^5 - 176*v^4 + 952*v^3 + 9769*v^2 - 24904*v - 146514) / 960 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 5\nu^{4} + 119\nu^{3} - 643\nu^{2} - 3370\nu + 16618 ) / 96$$ (-v^5 + 5*v^4 + 119*v^3 - 643*v^2 - 3370*v + 16618) / 96 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 10\nu^{5} + 218\nu^{4} + 1094\nu^{3} - 13231\nu^{2} - 21356\nu + 185766 ) / 960$$ (-v^6 - 10*v^5 + 218*v^4 + 1094*v^3 - 13231*v^2 - 21356*v + 185766) / 960 $$\beta_{5}$$ $$=$$ $$( 2\nu^{6} - 11\nu^{5} - 257\nu^{4} + 1069\nu^{3} + 8713\nu^{2} - 15198\nu - 90278 ) / 480$$ (2*v^6 - 11*v^5 - 257*v^4 + 1069*v^3 + 8713*v^2 - 15198*v - 90278) / 480 $$\beta_{6}$$ $$=$$ $$( \nu^{6} - 20\nu^{5} + 12\nu^{4} + 2316\nu^{3} - 10299\nu^{2} - 52544\nu + 221094 ) / 320$$ (v^6 - 20*v^5 + 12*v^4 + 2316*v^3 - 10299*v^2 - 52544*v + 221094) / 320
 $$\nu$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta _1 + 7 ) / 16$$ (-b3 - b2 + b1 + 7) / 16 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - \beta_{5} + 3\beta_{4} - 7\beta_{3} + 7\beta_{2} - 3\beta _1 + 861 ) / 16$$ (b6 - b5 + 3*b4 - 7*b3 + 7*b2 - 3*b1 + 861) / 16 $$\nu^{3}$$ $$=$$ $$( 13\beta_{6} - 29\beta_{5} - 41\beta_{4} - 75\beta_{3} - 37\beta_{2} + 73\beta _1 - 151 ) / 16$$ (13*b6 - 29*b5 - 41*b4 - 75*b3 - 37*b2 + 73*b1 - 151) / 16 $$\nu^{4}$$ $$=$$ $$( 143\beta_{6} - 111\beta_{5} + 269\beta_{4} - 757\beta_{3} + 637\beta_{2} - 353\beta _1 + 61287 ) / 16$$ (143*b6 - 111*b5 + 269*b4 - 757*b3 + 637*b2 - 353*b1 + 61287) / 16 $$\nu^{5}$$ $$=$$ $$( 1619\beta_{6} - 3363\beta_{5} - 5463\beta_{4} - 6375\beta_{3} - 2349\beta_{2} + 5481\beta _1 - 22859 ) / 16$$ (1619*b6 - 3363*b5 - 5463*b4 - 6375*b3 - 2349*b2 + 5481*b1 - 22859) / 16 $$\nu^{6}$$ $$=$$ $$( 15975\beta_{6} - 9063\beta_{5} + 13365\beta_{4} - 69353\beta_{3} + 50617\beta_{2} - 33565\beta _1 + 4854835 ) / 16$$ (15975*b6 - 9063*b5 + 13365*b4 - 69353*b3 + 50617*b2 - 33565*b1 + 4854835) / 16

## 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
 4.90786 9.56883 −4.83960 −8.92709 3.60554 −9.15396 7.83842
0 −29.3989 0 64.0801 0 138.793 0 621.298 0
1.2 0 −18.3274 0 −84.3249 0 −216.816 0 92.8952 0
1.3 0 −15.9679 0 31.5616 0 −106.304 0 11.9736 0
1.4 0 −15.4219 0 −58.0818 0 210.388 0 −5.16616 0
1.5 0 13.2844 0 69.0035 0 −156.573 0 −66.5241 0
1.6 0 15.2661 0 64.0682 0 −91.1564 0 −9.94539 0
1.7 0 24.5656 0 −54.3066 0 37.6697 0 360.469 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$29$$ $$-1$$

## 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 464.6.a.k 7
4.b odd 2 1 29.6.a.b 7
12.b even 2 1 261.6.a.e 7
20.d odd 2 1 725.6.a.b 7
116.d odd 2 1 841.6.a.b 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.b 7 4.b odd 2 1
261.6.a.e 7 12.b even 2 1
464.6.a.k 7 1.a even 1 1 trivial
725.6.a.b 7 20.d odd 2 1
841.6.a.b 7 116.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{7} + 26T_{3}^{6} - 1015T_{3}^{5} - 26056T_{3}^{4} + 280279T_{3}^{3} + 7496290T_{3}^{2} - 22844001T_{3} - 661023756$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(464))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7}$$
$3$ $$T^{7} + 26 T^{6} + \cdots - 661023756$$
$5$ $$T^{7} - 32 T^{6} + \cdots + 2378174390186$$
$7$ $$T^{7} + \cdots - 361848785235968$$
$11$ $$T^{7} + 1106 T^{6} + \cdots - 51\!\cdots\!44$$
$13$ $$T^{7} - 408 T^{6} + \cdots + 10\!\cdots\!34$$
$17$ $$T^{7} + 874 T^{6} + \cdots - 17\!\cdots\!28$$
$19$ $$T^{7} + 4288 T^{6} + \cdots - 15\!\cdots\!92$$
$23$ $$T^{7} - 4532 T^{6} + \cdots + 15\!\cdots\!56$$
$29$ $$(T - 841)^{7}$$
$31$ $$T^{7} + 7794 T^{6} + \cdots - 64\!\cdots\!48$$
$37$ $$T^{7} - 5086 T^{6} + \cdots - 16\!\cdots\!56$$
$41$ $$T^{7} - 19826 T^{6} + \cdots - 56\!\cdots\!72$$
$43$ $$T^{7} + 19498 T^{6} + \cdots + 58\!\cdots\!60$$
$47$ $$T^{7} + 14278 T^{6} + \cdots + 14\!\cdots\!36$$
$53$ $$T^{7} + 58644 T^{6} + \cdots + 84\!\cdots\!94$$
$59$ $$T^{7} + 12888 T^{6} + \cdots - 15\!\cdots\!00$$
$61$ $$T^{7} - 102866 T^{6} + \cdots + 45\!\cdots\!80$$
$67$ $$T^{7} + 102996 T^{6} + \cdots + 11\!\cdots\!60$$
$71$ $$T^{7} - 51596 T^{6} + \cdots + 20\!\cdots\!52$$
$73$ $$T^{7} + 17566 T^{6} + \cdots + 43\!\cdots\!96$$
$79$ $$T^{7} + 212058 T^{6} + \cdots + 28\!\cdots\!00$$
$83$ $$T^{7} - 122928 T^{6} + \cdots + 97\!\cdots\!52$$
$89$ $$T^{7} + 66510 T^{6} + \cdots - 79\!\cdots\!20$$
$97$ $$T^{7} + 118182 T^{6} + \cdots + 62\!\cdots\!52$$