# Properties

 Label 2312.4.a.k Level $2312$ Weight $4$ Character orbit 2312.a Self dual yes Analytic conductor $136.412$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2312 = 2^{3} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2312.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$136.412415933$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152$$ x^8 - 95*x^6 + 756*x^4 - 1780*x^2 + 1152 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: no (minimal twist has level 136) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + 16) q^{9}+O(q^{10})$$ q - b2 * q^3 + b4 * q^5 + (-b4 - b2 - b1) * q^7 + (-b5 + 16) * q^9 $$q - \beta_{2} q^{3} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + 16) q^{9} + (\beta_{7} + \beta_{4} - 2 \beta_{2}) q^{11} + (\beta_{6} + 5) q^{13} + (\beta_{6} + \beta_{3} - 3) q^{15} + ( - \beta_{6} + \beta_{3} - 5) q^{19} + ( - 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + 38) q^{21} + (\beta_{7} - 5 \beta_1) q^{23} + (\beta_{6} - 3 \beta_{5} + 63) q^{25} + (3 \beta_{7} - 3 \beta_{4} - 19 \beta_{2} - 6 \beta_1) q^{27} + (2 \beta_{7} - 5 \beta_{4} - 18 \beta_{2}) q^{29} + ( - 3 \beta_{7} + 4 \beta_{4} - 14 \beta_{2} - 5 \beta_1) q^{31} + (3 \beta_{6} - 5 \beta_{5} + \beta_{3} + 98) q^{33} + (2 \beta_{6} + 8 \beta_{5} - 130) q^{35} + ( - \beta_{4} + 28 \beta_{2} - 4 \beta_1) q^{37} + (3 \beta_{7} + 21 \beta_{4} - 3 \beta_{2} + 10 \beta_1) q^{39} + ( - 6 \beta_{7} - 16 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{41} + (\beta_{6} - 4 \beta_{5} + \beta_{3} - 3) q^{43} + (2 \beta_{7} + 43 \beta_{4} + 10 \beta_{2} + 20 \beta_1) q^{45} + ( - 3 \beta_{6} + \beta_{3} + 41) q^{47} + ( - \beta_{6} - 6 \beta_{5} - 4 \beta_{3} + 136) q^{49} + (3 \beta_{6} + 10 \beta_{5} - 3 \beta_{3} - 57) q^{53} + (7 \beta_{6} - 8 \beta_{5} + 3 \beta_{3} + 171) q^{55} + ( - 4 \beta_{7} + 28 \beta_{4} + 8 \beta_{2}) q^{57} + ( - \beta_{6} + 4 \beta_{5} - \beta_{3} + 11) q^{59} + ( - 6 \beta_{7} + 5 \beta_{4} - 74 \beta_{2} - 16 \beta_1) q^{61} + (\beta_{7} - 70 \beta_{4} - 80 \beta_{2} - 15 \beta_1) q^{63} + (10 \beta_{7} + 16 \beta_{4} - 78 \beta_{2} - 20 \beta_1) q^{65} + ( - 9 \beta_{6} + 4 \beta_{5} - 3 \beta_{3} - 73) q^{67} + ( - 3 \beta_{6} - 8 \beta_{5} - 25) q^{69} + (6 \beta_{7} - 27 \beta_{4} + 31 \beta_{2} - 29 \beta_1) q^{71} + ( - 6 \beta_{7} + 26 \beta_{4} + 6 \beta_{2} - 20 \beta_1) q^{73} + (12 \beta_{7} + 12 \beta_{4} - 151 \beta_{2} - 8 \beta_1) q^{75} + ( - 3 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - 207) q^{77} + ( - 11 \beta_{7} - 16 \beta_{4} - 64 \beta_{2} - 5 \beta_1) q^{79} + ( - 3 \beta_{6} - 7 \beta_{5} - 3 \beta_{3} + 391) q^{81} + ( - 9 \beta_{6} + 16 \beta_{5} - \beta_{3} - 297) q^{83} + ( - \beta_{6} - 24 \beta_{5} - 5 \beta_{3} + 819) q^{87} + (\beta_{6} + 18 \beta_{5} + 4 \beta_{3} + 19) q^{89} + (\beta_{7} + 21 \beta_{4} + 139 \beta_{2} - 20 \beta_1) q^{91} + ( - 7 \beta_{6} - 10 \beta_{5} + 4 \beta_{3} + 505) q^{93} + ( - 8 \beta_{7} - 12 \beta_{4} - 120 \beta_{2} + 32 \beta_1) q^{95} + ( - 16 \beta_{7} + 22 \beta_{4} - 140 \beta_{2} + 24 \beta_1) q^{97} + ( - 4 \beta_{7} + 70 \beta_{4} - 183 \beta_{2} + 10 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^3 + b4 * q^5 + (-b4 - b2 - b1) * q^7 + (-b5 + 16) * q^9 + (b7 + b4 - 2*b2) * q^11 + (b6 + 5) * q^13 + (b6 + b3 - 3) * q^15 + (-b6 + b3 - 5) * q^19 + (-2*b6 - 2*b5 - b3 + 38) * q^21 + (b7 - 5*b1) * q^23 + (b6 - 3*b5 + 63) * q^25 + (3*b7 - 3*b4 - 19*b2 - 6*b1) * q^27 + (2*b7 - 5*b4 - 18*b2) * q^29 + (-3*b7 + 4*b4 - 14*b2 - 5*b1) * q^31 + (3*b6 - 5*b5 + b3 + 98) * q^33 + (2*b6 + 8*b5 - 130) * q^35 + (-b4 + 28*b2 - 4*b1) * q^37 + (3*b7 + 21*b4 - 3*b2 + 10*b1) * q^39 + (-6*b7 - 16*b4 + 2*b2 + 4*b1) * q^41 + (b6 - 4*b5 + b3 - 3) * q^43 + (2*b7 + 43*b4 + 10*b2 + 20*b1) * q^45 + (-3*b6 + b3 + 41) * q^47 + (-b6 - 6*b5 - 4*b3 + 136) * q^49 + (3*b6 + 10*b5 - 3*b3 - 57) * q^53 + (7*b6 - 8*b5 + 3*b3 + 171) * q^55 + (-4*b7 + 28*b4 + 8*b2) * q^57 + (-b6 + 4*b5 - b3 + 11) * q^59 + (-6*b7 + 5*b4 - 74*b2 - 16*b1) * q^61 + (b7 - 70*b4 - 80*b2 - 15*b1) * q^63 + (10*b7 + 16*b4 - 78*b2 - 20*b1) * q^65 + (-9*b6 + 4*b5 - 3*b3 - 73) * q^67 + (-3*b6 - 8*b5 - 25) * q^69 + (6*b7 - 27*b4 + 31*b2 - 29*b1) * q^71 + (-6*b7 + 26*b4 + 6*b2 - 20*b1) * q^73 + (12*b7 + 12*b4 - 151*b2 - 8*b1) * q^75 + (-3*b6 + 2*b5 - 4*b3 - 207) * q^77 + (-11*b7 - 16*b4 - 64*b2 - 5*b1) * q^79 + (-3*b6 - 7*b5 - 3*b3 + 391) * q^81 + (-9*b6 + 16*b5 - b3 - 297) * q^83 + (-b6 - 24*b5 - 5*b3 + 819) * q^87 + (b6 + 18*b5 + 4*b3 + 19) * q^89 + (b7 + 21*b4 + 139*b2 - 20*b1) * q^91 + (-7*b6 - 10*b5 + 4*b3 + 505) * q^93 + (-8*b7 - 12*b4 - 120*b2 + 32*b1) * q^95 + (-16*b7 + 22*b4 - 140*b2 + 24*b1) * q^97 + (-4*b7 + 70*b4 - 183*b2 + 10*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 132 q^{9}+O(q^{10})$$ 8 * q + 132 * q^9 $$8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100})$$ 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu$$ 4*v $$\beta_{2}$$ $$=$$ $$( 3\nu^{7} - 284\nu^{5} + 2159\nu^{3} - 3402\nu ) / 172$$ (3*v^7 - 284*v^5 + 2159*v^3 - 3402*v) / 172 $$\beta_{3}$$ $$=$$ $$( 35\nu^{6} - 3127\nu^{4} + 8318\nu^{2} + 12168 ) / 172$$ (35*v^6 - 3127*v^4 + 8318*v^2 + 12168) / 172 $$\beta_{4}$$ $$=$$ $$( -59\nu^{7} + 5485\nu^{5} - 33588\nu^{3} + 47900\nu ) / 1032$$ (-59*v^7 + 5485*v^5 - 33588*v^3 + 47900*v) / 1032 $$\beta_{5}$$ $$=$$ $$( 41\nu^{6} - 3781\nu^{4} + 20634\nu^{2} - 22844 ) / 172$$ (41*v^6 - 3781*v^4 + 20634*v^2 - 22844) / 172 $$\beta_{6}$$ $$=$$ $$( -45\nu^{6} + 4217\nu^{4} - 28386\nu^{2} + 35292 ) / 172$$ (-45*v^6 + 4217*v^4 - 28386*v^2 + 35292) / 172 $$\beta_{7}$$ $$=$$ $$( 109\nu^{7} - 9989\nu^{5} + 48702\nu^{3} - 15160\nu ) / 1032$$ (109*v^7 - 9989*v^5 + 48702*v^3 - 15160*v) / 1032
 $$\nu$$ $$=$$ $$( \beta_1 ) / 4$$ (b1) / 4 $$\nu^{2}$$ $$=$$ $$( 3\beta_{6} + 5\beta_{5} - 2\beta_{3} + 190 ) / 8$$ (3*b6 + 5*b5 - 2*b3 + 190) / 8 $$\nu^{3}$$ $$=$$ $$( -7\beta_{7} - 23\beta_{4} - 33\beta_{2} + 78\beta_1 ) / 4$$ (-7*b7 - 23*b4 - 33*b2 + 78*b1) / 4 $$\nu^{4}$$ $$=$$ $$( 277\beta_{6} + 450\beta_{5} - 171\beta_{3} + 15027 ) / 8$$ (277*b6 + 450*b5 - 171*b3 + 15027) / 8 $$\nu^{5}$$ $$=$$ $$( -619\beta_{7} - 2075\beta_{4} - 3053\beta_{2} + 6708\beta_1 ) / 4$$ (-619*b7 - 2075*b4 - 3053*b2 + 6708*b1) / 4 $$\nu^{6}$$ $$=$$ $$( 24035\beta_{6} + 39016\beta_{5} - 14763\beta_{3} + 1294619 ) / 8$$ (24035*b6 + 39016*b5 - 14763*b3 + 1294619) / 8 $$\nu^{7}$$ $$=$$ $$( -53561\beta_{7} - 179881\beta_{4} - 265039\beta_{2} + 580024\beta_1 ) / 4$$ (-53561*b7 - 179881*b4 - 265039*b2 + 580024*b1) / 4

## 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
 2.20783 −1.03229 1.60125 −9.30031 9.30031 −1.60125 1.03229 −2.20783
0 −9.26065 0 16.4090 0 −34.5010 0 58.7597 0
1.2 0 −8.52350 0 −18.2701 0 13.8757 0 45.6501 0
1.3 0 −2.95309 0 −4.89575 0 −4.46235 0 −18.2793 0
1.4 0 −2.62097 0 11.5318 0 23.0485 0 −20.1305 0
1.5 0 2.62097 0 −11.5318 0 −23.0485 0 −20.1305 0
1.6 0 2.95309 0 4.89575 0 4.46235 0 −18.2793 0
1.7 0 8.52350 0 18.2701 0 −13.8757 0 45.6501 0
1.8 0 9.26065 0 −16.4090 0 34.5010 0 58.7597 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$17$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2312.4.a.k 8
17.b even 2 1 inner 2312.4.a.k 8
17.c even 4 2 136.4.b.b 8
51.f odd 4 2 1224.4.c.e 8
68.f odd 4 2 272.4.b.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.b 8 17.c even 4 2
272.4.b.f 8 68.f odd 4 2
1224.4.c.e 8 51.f odd 4 2
2312.4.a.k 8 1.a even 1 1 trivial
2312.4.a.k 8 17.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 174T_{3}^{6} + 8760T_{3}^{4} - 106624T_{3}^{2} + 373248$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2312))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 174 T^{6} + 8760 T^{4} + \cdots + 373248$$
$5$ $$T^{8} - 760 T^{6} + \cdots + 286466048$$
$7$ $$T^{8} - 1934 T^{6} + \cdots + 2424307712$$
$11$ $$T^{8} - 7406 T^{6} + \cdots + 1063158272$$
$13$ $$(T^{4} - 22 T^{3} - 5836 T^{2} + \cdots + 8525216)^{2}$$
$17$ $$T^{8}$$
$19$ $$(T^{4} + 24 T^{3} - 21056 T^{2} + \cdots + 44946176)^{2}$$
$23$ $$T^{8} - 36734 T^{6} + \cdots + 2935871578112$$
$29$ $$T^{8} - 106232 T^{6} + \cdots + 14\!\cdots\!52$$
$31$ $$T^{8} - 155918 T^{6} + \cdots + 32\!\cdots\!52$$
$37$ $$T^{8} - 166072 T^{6} + \cdots + 54\!\cdots\!12$$
$41$ $$T^{8} - 419392 T^{6} + \cdots + 55\!\cdots\!32$$
$43$ $$(T^{4} + 4 T^{3} - 60592 T^{2} + \cdots + 112195072)^{2}$$
$47$ $$(T^{4} - 156 T^{3} - 57344 T^{2} + \cdots + 134217728)^{2}$$
$53$ $$(T^{4} + 236 T^{3} + \cdots - 14761769616)^{2}$$
$59$ $$(T^{4} - 36 T^{3} - 60112 T^{2} + \cdots + 70465536)^{2}$$
$61$ $$T^{8} - 1597240 T^{6} + \cdots + 18\!\cdots\!28$$
$67$ $$(T^{4} + 312 T^{3} + \cdots + 30967766784)^{2}$$
$71$ $$T^{8} - 1607086 T^{6} + \cdots + 14\!\cdots\!28$$
$73$ $$T^{8} - 1779648 T^{6} + \cdots + 21\!\cdots\!68$$
$79$ $$T^{8} - 1578638 T^{6} + \cdots + 17\!\cdots\!72$$
$83$ $$(T^{4} + 1236 T^{3} + \cdots - 54225864704)^{2}$$
$89$ $$(T^{4} - 34 T^{3} - 1202908 T^{2} + \cdots + 21605388512)^{2}$$
$97$ $$T^{8} - 5185024 T^{6} + \cdots + 13\!\cdots\!92$$