# Properties

 Label 464.6.a.i Level $464$ Weight $6$ Character orbit 464.a Self dual yes Analytic conductor $74.418$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.3257317.1 Defining polynomial: $$x^{4} - 34x^{2} - 27x + 10$$ x^4 - 34*x^2 - 27*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 7) q^{3} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 17) q^{5} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 52) q^{7} + ( - 14 \beta_{2} + 3 \beta_1 - 70) q^{9}+O(q^{10})$$ q + (-b2 + 7) * q^3 + (b3 + 2*b2 - 2*b1 - 17) * q^5 + (3*b3 + 2*b2 - b1 + 52) * q^7 + (-14*b2 + 3*b1 - 70) * q^9 $$q + ( - \beta_{2} + 7) q^{3} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 17) q^{5} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 52) q^{7} + ( - 14 \beta_{2} + 3 \beta_1 - 70) q^{9} + (5 \beta_{3} + 25 \beta_{2} - 2 \beta_1 + 31) q^{11} + (15 \beta_{3} - 24 \beta_{2} + 7 \beta_1 - 115) q^{13} + (20 \beta_{3} + 39 \beta_{2} + 5 \beta_1 - 233) q^{15} + ( - \beta_{3} + 32 \beta_{2} - 9 \beta_1 + 46) q^{17} + ( - 29 \beta_{3} - 44 \beta_{2} - 62 \beta_1 + 598) q^{19} + (55 \beta_{3} - 24 \beta_{2} + 7 \beta_1 + 248) q^{21} + (35 \beta_{3} - 80 \beta_{2} - 33 \beta_1 + 298) q^{23} + ( - 70 \beta_{3} - 180 \beta_{2} + 20 \beta_1 + 456) q^{25} + ( - 3 \beta_{3} + 209 \beta_{2} + 30 \beta_1 - 617) q^{27} - 841 q^{29} + (23 \beta_{3} - 83 \beta_{2} - 201 \beta_1 + 4803) q^{31} + (92 \beta_{3} + 168 \beta_{2} - 52 \beta_1 - 2645) q^{33} + ( - 55 \beta_{3} - 434 \beta_{2} - 65 \beta_1 + 5736) q^{35} + (10 \beta_{3} - 84 \beta_{2} - 314 \beta_1 - 2732) q^{37} + (263 \beta_{3} - 7 \beta_{2} + 89 \beta_1 + 2183) q^{39} + ( - 201 \beta_{3} + 792 \beta_{2} - 579 \beta_1 - 280) q^{41} + ( - 193 \beta_{3} - 631 \beta_{2} + 486 \beta_1 + 5355) q^{43} + (112 \beta_{3} + 90 \beta_{2} + 409 \beta_1 - 2086) q^{45} + ( - 18 \beta_{3} + 613 \beta_{2} + 531 \beta_1 - 5943) q^{47} + (260 \beta_{3} - 1288 \beta_{2} + 68 \beta_1 + 2613) q^{49} + ( - 9 \beta_{3} + 192 \beta_{2} - 63 \beta_1 - 3186) q^{51} + (157 \beta_{3} - 180 \beta_{2} + 1039 \beta_1 + 2215) q^{53} + ( - 429 \beta_{3} - 1363 \beta_{2} - 409 \beta_1 + 13163) q^{55} + ( - 460 \beta_{3} - 898 \beta_{2} + 293 \beta_1 + 12236) q^{57} + (35 \beta_{3} - 316 \beta_{2} - 1391 \beta_1 + 2710) q^{59} + ( - 599 \beta_{3} + 220 \beta_{2} - 1681 \beta_1 + 12362) q^{61} + (254 \beta_{3} - 696 \beta_{2} + 452 \beta_1 - 6872) q^{63} + ( - 208 \beta_{3} - 2550 \beta_{2} + 1083 \beta_1 + 24459) q^{65} + (258 \beta_{3} - 124 \beta_{2} + 2276 \beta_1 + 1960) q^{67} + (663 \beta_{3} - 652 \beta_{2} + 477 \beta_1 + 14698) q^{69} + (24 \beta_{3} + 174 \beta_{2} - 1712 \beta_1 + 12186) q^{71} + ( - 376 \beta_{3} + 792 \beta_{2} + 1496 \beta_1 - 18748) q^{73} + ( - 1280 \beta_{3} - 2036 \beta_{2} + 250 \beta_1 + 22612) q^{75} + ( - 469 \beta_{3} - 1432 \beta_{2} - 155 \beta_1 + 32164) q^{77} + ( - 716 \beta_{3} + 3679 \beta_{2} - 1111 \beta_1 + 26519) q^{79} + ( - 84 \beta_{3} + 5410 \beta_{2} - 1485 \beta_1 - 14923) q^{81} + (277 \beta_{3} + 1236 \beta_{2} + 1901 \beta_1 - 15722) q^{83} + ( - 343 \beta_{3} - 172 \beta_{2} - 735 \beta_1 + 5962) q^{85} + (841 \beta_{2} - 5887) q^{87} + (1095 \beta_{3} - 4112 \beta_{2} + 1635 \beta_1 + 26892) q^{89} + (1283 \beta_{3} - 8578 \beta_{2} + 1935 \beta_1 + 67224) q^{91} + (615 \beta_{3} - 4890 \beta_{2} + 1122 \beta_1 + 55365) q^{93} + (1953 \beta_{3} + 9880 \beta_{2} - 1176 \beta_1 - 36838) q^{95} + (1795 \beta_{3} + 5556 \beta_{2} - 797 \beta_1 - 12380) q^{97} + (493 \beta_{3} - 1782 \beta_{2} + 466 \beta_1 - 41680) q^{99}+O(q^{100})$$ q + (-b2 + 7) * q^3 + (b3 + 2*b2 - 2*b1 - 17) * q^5 + (3*b3 + 2*b2 - b1 + 52) * q^7 + (-14*b2 + 3*b1 - 70) * q^9 + (5*b3 + 25*b2 - 2*b1 + 31) * q^11 + (15*b3 - 24*b2 + 7*b1 - 115) * q^13 + (20*b3 + 39*b2 + 5*b1 - 233) * q^15 + (-b3 + 32*b2 - 9*b1 + 46) * q^17 + (-29*b3 - 44*b2 - 62*b1 + 598) * q^19 + (55*b3 - 24*b2 + 7*b1 + 248) * q^21 + (35*b3 - 80*b2 - 33*b1 + 298) * q^23 + (-70*b3 - 180*b2 + 20*b1 + 456) * q^25 + (-3*b3 + 209*b2 + 30*b1 - 617) * q^27 - 841 * q^29 + (23*b3 - 83*b2 - 201*b1 + 4803) * q^31 + (92*b3 + 168*b2 - 52*b1 - 2645) * q^33 + (-55*b3 - 434*b2 - 65*b1 + 5736) * q^35 + (10*b3 - 84*b2 - 314*b1 - 2732) * q^37 + (263*b3 - 7*b2 + 89*b1 + 2183) * q^39 + (-201*b3 + 792*b2 - 579*b1 - 280) * q^41 + (-193*b3 - 631*b2 + 486*b1 + 5355) * q^43 + (112*b3 + 90*b2 + 409*b1 - 2086) * q^45 + (-18*b3 + 613*b2 + 531*b1 - 5943) * q^47 + (260*b3 - 1288*b2 + 68*b1 + 2613) * q^49 + (-9*b3 + 192*b2 - 63*b1 - 3186) * q^51 + (157*b3 - 180*b2 + 1039*b1 + 2215) * q^53 + (-429*b3 - 1363*b2 - 409*b1 + 13163) * q^55 + (-460*b3 - 898*b2 + 293*b1 + 12236) * q^57 + (35*b3 - 316*b2 - 1391*b1 + 2710) * q^59 + (-599*b3 + 220*b2 - 1681*b1 + 12362) * q^61 + (254*b3 - 696*b2 + 452*b1 - 6872) * q^63 + (-208*b3 - 2550*b2 + 1083*b1 + 24459) * q^65 + (258*b3 - 124*b2 + 2276*b1 + 1960) * q^67 + (663*b3 - 652*b2 + 477*b1 + 14698) * q^69 + (24*b3 + 174*b2 - 1712*b1 + 12186) * q^71 + (-376*b3 + 792*b2 + 1496*b1 - 18748) * q^73 + (-1280*b3 - 2036*b2 + 250*b1 + 22612) * q^75 + (-469*b3 - 1432*b2 - 155*b1 + 32164) * q^77 + (-716*b3 + 3679*b2 - 1111*b1 + 26519) * q^79 + (-84*b3 + 5410*b2 - 1485*b1 - 14923) * q^81 + (277*b3 + 1236*b2 + 1901*b1 - 15722) * q^83 + (-343*b3 - 172*b2 - 735*b1 + 5962) * q^85 + (841*b2 - 5887) * q^87 + (1095*b3 - 4112*b2 + 1635*b1 + 26892) * q^89 + (1283*b3 - 8578*b2 + 1935*b1 + 67224) * q^91 + (615*b3 - 4890*b2 + 1122*b1 + 55365) * q^93 + (1953*b3 + 9880*b2 - 1176*b1 - 36838) * q^95 + (1795*b3 + 5556*b2 - 797*b1 - 12380) * q^97 + (493*b3 - 1782*b2 + 466*b1 - 41680) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10})$$ 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9 $$4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9} + 124 q^{11} - 460 q^{13} - 932 q^{15} + 184 q^{17} + 2392 q^{19} + 992 q^{21} + 1192 q^{23} + 1824 q^{25} - 2468 q^{27} - 3364 q^{29} + 19212 q^{31} - 10580 q^{33} + 22944 q^{35} - 10928 q^{37} + 8732 q^{39} - 1120 q^{41} + 21420 q^{43} - 8344 q^{45} - 23772 q^{47} + 10452 q^{49} - 12744 q^{51} + 8860 q^{53} + 52652 q^{55} + 48944 q^{57} + 10840 q^{59} + 49448 q^{61} - 27488 q^{63} + 97836 q^{65} + 7840 q^{67} + 58792 q^{69} + 48744 q^{71} - 74992 q^{73} + 90448 q^{75} + 128656 q^{77} + 106076 q^{79} - 59692 q^{81} - 62888 q^{83} + 23848 q^{85} - 23548 q^{87} + 107568 q^{89} + 268896 q^{91} + 221460 q^{93} - 147352 q^{95} - 49520 q^{97} - 166720 q^{99}+O(q^{100})$$ 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9 + 124 * q^11 - 460 * q^13 - 932 * q^15 + 184 * q^17 + 2392 * q^19 + 992 * q^21 + 1192 * q^23 + 1824 * q^25 - 2468 * q^27 - 3364 * q^29 + 19212 * q^31 - 10580 * q^33 + 22944 * q^35 - 10928 * q^37 + 8732 * q^39 - 1120 * q^41 + 21420 * q^43 - 8344 * q^45 - 23772 * q^47 + 10452 * q^49 - 12744 * q^51 + 8860 * q^53 + 52652 * q^55 + 48944 * q^57 + 10840 * q^59 + 49448 * q^61 - 27488 * q^63 + 97836 * q^65 + 7840 * q^67 + 58792 * q^69 + 48744 * q^71 - 74992 * q^73 + 90448 * q^75 + 128656 * q^77 + 106076 * q^79 - 59692 * q^81 - 62888 * q^83 + 23848 * q^85 - 23548 * q^87 + 107568 * q^89 + 268896 * q^91 + 221460 * q^93 - 147352 * q^95 - 49520 * q^97 - 166720 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 34x^{2} - 27x + 10$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu$$ 4*v $$\beta_{2}$$ $$=$$ $$( 2\nu^{2} - 34 ) / 3$$ (2*v^2 - 34) / 3 $$\beta_{3}$$ $$=$$ $$( 8\nu^{3} - 4\nu^{2} - 268\nu - 94 ) / 3$$ (8*v^3 - 4*v^2 - 268*v - 94) / 3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 4$$ (b1) / 4 $$\nu^{2}$$ $$=$$ $$( 3\beta_{2} + 34 ) / 2$$ (3*b2 + 34) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} + 6\beta_{2} + 67\beta _1 + 162 ) / 8$$ (3*b3 + 6*b2 + 67*b1 + 162) / 8

## 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
 6.17343 −5.34807 −1.10057 0.275208
0 −7.07413 0 −44.4758 0 36.7447 0 −192.957 0
1.2 0 −0.734546 0 41.6400 0 90.0205 0 −242.460 0
1.3 0 17.5258 0 32.5670 0 220.793 0 64.1549 0
1.4 0 18.2828 0 −97.7313 0 −139.558 0 91.2623 0
 $$n$$: e.g. 2-40 or 990-1000 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.i 4
4.b odd 2 1 29.6.a.a 4
12.b even 2 1 261.6.a.a 4
20.d odd 2 1 725.6.a.a 4
116.d odd 2 1 841.6.a.a 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 28T_{3}^{3} + 46T_{3}^{2} + 2316T_{3} + 1665$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(464))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 28 T^{3} + 46 T^{2} + \cdots + 1665$$
$5$ $$T^{4} + 68 T^{3} - 4850 T^{2} + \cdots + 5894489$$
$7$ $$T^{4} - 208 T^{3} + \cdots - 101924272$$
$11$ $$T^{4} - 124 T^{3} + \cdots - 3717303119$$
$13$ $$T^{4} + 460 T^{3} + \cdots - 116053863479$$
$17$ $$T^{4} - 184 T^{3} + \cdots + 11464717824$$
$19$ $$T^{4} - 2392 T^{3} + \cdots + 2620094791680$$
$23$ $$T^{4} - 1192 T^{3} + \cdots + 2033080361984$$
$29$ $$(T + 841)^{4}$$
$31$ $$T^{4} + \cdots - 421127233952247$$
$37$ $$T^{4} + 10928 T^{3} + \cdots + 16079593861120$$
$41$ $$T^{4} + 1120 T^{3} + \cdots + 10\!\cdots\!56$$
$43$ $$T^{4} - 21420 T^{3} + \cdots + 47\!\cdots\!37$$
$47$ $$T^{4} + 23772 T^{3} + \cdots - 35\!\cdots\!87$$
$53$ $$T^{4} - 8860 T^{3} + \cdots + 16\!\cdots\!53$$
$59$ $$T^{4} - 10840 T^{3} + \cdots - 43\!\cdots\!80$$
$61$ $$T^{4} - 49448 T^{3} + \cdots + 90\!\cdots\!24$$
$67$ $$T^{4} - 7840 T^{3} + \cdots + 20\!\cdots\!92$$
$71$ $$T^{4} - 48744 T^{3} + \cdots - 18\!\cdots\!16$$
$73$ $$T^{4} + 74992 T^{3} + \cdots - 40\!\cdots\!00$$
$79$ $$T^{4} - 106076 T^{3} + \cdots - 53\!\cdots\!75$$
$83$ $$T^{4} + 62888 T^{3} + \cdots - 18\!\cdots\!88$$
$89$ $$T^{4} - 107568 T^{3} + \cdots - 18\!\cdots\!60$$
$97$ $$T^{4} + 49520 T^{3} + \cdots - 12\!\cdots\!28$$