# Properties

 Label 4368.2.h.s Level $4368$ Weight $2$ Character orbit 4368.h Analytic conductor $34.879$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.8786556029$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16$$ x^10 + 21*x^8 + 124*x^6 + 212*x^4 + 116*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 2184) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta_{6} - \beta_1) q^{5} - \beta_1 q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b6 - b1) * q^5 - b1 * q^7 + q^9 $$q - q^{3} + ( - \beta_{6} - \beta_1) q^{5} - \beta_1 q^{7} + q^{9} + ( - \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{3}) q^{13} + (\beta_{6} + \beta_1) q^{15} + ( - \beta_{5} + \beta_{2} - 2) q^{17} + ( - \beta_{9} + 2 \beta_1) q^{19} + \beta_1 q^{21} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{23} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 1) q^{25} - q^{27} + ( - \beta_{5} + \beta_{2}) q^{29} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{33} + ( - \beta_{2} - 1) q^{35} + ( - \beta_{8} - 2 \beta_{6}) q^{37} + (\beta_{5} - \beta_{3}) q^{39} + (\beta_{8} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{41} + (\beta_{7} - 2 \beta_{2}) q^{43} + ( - \beta_{6} - \beta_1) q^{45} + (\beta_{8} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{47} - q^{49} + (\beta_{5} - \beta_{2} + 2) q^{51} + (\beta_{7} - \beta_{5} - \beta_{2}) q^{53} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 2) q^{55} + (\beta_{9} - 2 \beta_1) q^{57} + (\beta_{9} + 3 \beta_{6} - \beta_{4} - \beta_{3} - \beta_1) q^{59} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{61} - \beta_1 q^{63} + ( - \beta_{9} + \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{3} + \beta_1 + 2) q^{65} + (\beta_{9} + \beta_{8} + 4 \beta_{6}) q^{67} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{69} + (\beta_{9} + 2 \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{71} + (\beta_{9} - \beta_{4} - \beta_{3} + 2 \beta_1) q^{73} + (\beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 1) q^{75} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{77} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - 4) q^{79} + q^{81} + ( - \beta_{9} - 2 \beta_{8} - 5 \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{83} + ( - \beta_{9} + 2 \beta_{6}) q^{85} + (\beta_{5} - \beta_{2}) q^{87} + (\beta_{8} + 5 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{89} + (\beta_{8} + \beta_{6} - \beta_{4}) q^{91} + ( - \beta_{7} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4) q^{95} + (2 \beta_{8} - \beta_{4} - \beta_{3} - 4 \beta_1) q^{97} + ( - \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q - q^3 + (-b6 - b1) * q^5 - b1 * q^7 + q^9 + (-b6 + b4 + b3 - b1) * q^11 + (-b5 + b3) * q^13 + (b6 + b1) * q^15 + (-b5 + b2 - 2) * q^17 + (-b9 + 2*b1) * q^19 + b1 * q^21 + (b5 + b4 - b3 + b2 + 2) * q^23 + (-b5 - b4 + b3 - 3*b2 - 1) * q^25 - q^27 + (-b5 + b2) * q^29 + (b6 - b4 - b3 + b1) * q^33 + (-b2 - 1) * q^35 + (-b8 - 2*b6) * q^37 + (b5 - b3) * q^39 + (b8 + b6 - 2*b4 - 2*b3 - b1) * q^41 + (b7 - 2*b2) * q^43 + (-b6 - b1) * q^45 + (b8 - b6 + b4 + b3 - b1) * q^47 - q^49 + (b5 - b2 + 2) * q^51 + (b7 - b5 - b2) * q^53 + (-b5 - b4 + b3 - 3*b2 - 2) * q^55 + (b9 - 2*b1) * q^57 + (b9 + 3*b6 - b4 - b3 - b1) * q^59 + (b7 - b4 + b3) * q^61 - b1 * q^63 + (-b9 + b8 + 3*b6 - b4 - b3 + b1 + 2) * q^65 + (b9 + b8 + 4*b6) * q^67 + (-b5 - b4 + b3 - b2 - 2) * q^69 + (b9 + 2*b8 + 3*b6 - b4 - b3 + b1) * q^71 + (b9 - b4 - b3 + 2*b1) * q^73 + (b5 + b4 - b3 + 3*b2 + 1) * q^75 + (-b4 + b3 - b2 - 1) * q^77 + (b7 + b5 + b4 - b3 + 3*b2 - 4) * q^79 + q^81 + (-b9 - 2*b8 - 5*b6 + b4 + b3 - b1) * q^83 + (-b9 + 2*b6) * q^85 + (b5 - b2) * q^87 + (b8 + 5*b6 - 2*b4 - 2*b3 - b1) * q^89 + (b8 + b6 - b4) * q^91 + (-b7 + 4*b5 + 2*b4 - 2*b3 + 2*b2 + 4) * q^95 + (2*b8 - b4 - b3 - 4*b1) * q^97 + (-b6 + b4 + b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 10 q^{3} + 10 q^{9}+O(q^{10})$$ 10 * q - 10 * q^3 + 10 * q^9 $$10 q - 10 q^{3} + 10 q^{9} - 22 q^{17} + 18 q^{23} - 8 q^{25} - 10 q^{27} - 2 q^{29} - 10 q^{35} + 2 q^{43} - 10 q^{49} + 22 q^{51} - 18 q^{55} + 6 q^{61} + 20 q^{65} - 18 q^{69} + 8 q^{75} - 6 q^{77} - 40 q^{79} + 10 q^{81} + 2 q^{87} + 2 q^{91} + 38 q^{95}+O(q^{100})$$ 10 * q - 10 * q^3 + 10 * q^9 - 22 * q^17 + 18 * q^23 - 8 * q^25 - 10 * q^27 - 2 * q^29 - 10 * q^35 + 2 * q^43 - 10 * q^49 + 22 * q^51 - 18 * q^55 + 6 * q^61 + 20 * q^65 - 18 * q^69 + 8 * q^75 - 6 * q^77 - 40 * q^79 + 10 * q^81 + 2 * q^87 + 2 * q^91 + 38 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -5\nu^{9} - 89\nu^{7} - 296\nu^{5} + 632\nu^{3} + 1004\nu ) / 392$$ (-5*v^9 - 89*v^7 - 296*v^5 + 632*v^3 + 1004*v) / 392 $$\beta_{2}$$ $$=$$ $$( -5\nu^{8} - 89\nu^{6} - 296\nu^{4} + 632\nu^{2} + 808 ) / 196$$ (-5*v^8 - 89*v^6 - 296*v^4 + 632*v^2 + 808) / 196 $$\beta_{3}$$ $$=$$ $$( -4\nu^{8} - 81\nu^{6} - 423\nu^{4} - 396\nu^{2} + 98\nu - 20 ) / 98$$ (-4*v^8 - 81*v^6 - 423*v^4 - 396*v^2 + 98*v - 20) / 98 $$\beta_{4}$$ $$=$$ $$( 4\nu^{8} + 81\nu^{6} + 423\nu^{4} + 396\nu^{2} + 98\nu + 20 ) / 98$$ (4*v^8 + 81*v^6 + 423*v^4 + 396*v^2 + 98*v + 20) / 98 $$\beta_{5}$$ $$=$$ $$( 12\nu^{8} + 243\nu^{6} + 1318\nu^{4} + 1727\nu^{2} + 501 ) / 49$$ (12*v^8 + 243*v^6 + 1318*v^4 + 1727*v^2 + 501) / 49 $$\beta_{6}$$ $$=$$ $$( 27\nu^{9} + 559\nu^{7} + 3186\nu^{5} + 4878\nu^{3} + 2340\nu ) / 196$$ (27*v^9 + 559*v^7 + 3186*v^5 + 4878*v^3 + 2340*v) / 196 $$\beta_{7}$$ $$=$$ $$( -65\nu^{8} - 1353\nu^{6} - 7768\nu^{4} - 11580\nu^{2} - 3020 ) / 196$$ (-65*v^8 - 1353*v^6 - 7768*v^4 - 11580*v^2 - 3020) / 196 $$\beta_{8}$$ $$=$$ $$( 65\nu^{9} + 1353\nu^{7} + 7768\nu^{5} + 11580\nu^{3} + 2628\nu ) / 392$$ (65*v^9 + 1353*v^7 + 7768*v^5 + 11580*v^3 + 2628*v) / 392 $$\beta_{9}$$ $$=$$ $$( -289\nu^{9} - 5889\nu^{7} - 32240\nu^{5} - 42380\nu^{3} - 11588\nu ) / 392$$ (-289*v^9 - 5889*v^7 - 32240*v^5 - 42380*v^3 - 11588*v) / 392
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} ) / 2$$ (b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 3\beta_{2} - 8 ) / 2$$ (b7 + b5 + 2*b4 - 2*b3 + 3*b2 - 8) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{9} - \beta_{8} + 6\beta_{6} - 10\beta_{4} - 10\beta_{3} - 6\beta_1 ) / 2$$ (b9 - b8 + 6*b6 - 10*b4 - 10*b3 - 6*b1) / 2 $$\nu^{4}$$ $$=$$ $$( -11\beta_{7} - 9\beta_{5} - 28\beta_{4} + 28\beta_{3} - 33\beta_{2} + 70 ) / 2$$ (-11*b7 - 9*b5 - 28*b4 + 28*b3 - 33*b2 + 70) / 2 $$\nu^{5}$$ $$=$$ $$( -19\beta_{9} + 3\beta_{8} - 94\beta_{6} + 114\beta_{4} + 114\beta_{3} + 122\beta_1 ) / 2$$ (-19*b9 + 3*b8 - 94*b6 + 114*b4 + 114*b3 + 122*b1) / 2 $$\nu^{6}$$ $$=$$ $$( 117\beta_{7} + 79\beta_{5} + 358\beta_{4} - 358\beta_{3} + 383\beta_{2} - 730 ) / 2$$ (117*b7 + 79*b5 + 358*b4 - 358*b3 + 383*b2 - 730) / 2 $$\nu^{7}$$ $$=$$ $$( 279\beta_{9} + 45\beta_{8} + 1274\beta_{6} - 1350\beta_{4} - 1350\beta_{3} - 1782\beta_1 ) / 2$$ (279*b9 + 45*b8 + 1274*b6 - 1350*b4 - 1350*b3 - 1782*b1) / 2 $$\nu^{8}$$ $$=$$ $$( -1305\beta_{7} - 747\beta_{5} - 4462\beta_{4} + 4462\beta_{3} - 4563\beta_{2} + 8162 ) / 2$$ (-1305*b7 - 747*b5 - 4462*b4 + 4462*b3 - 4563*b2 + 8162) / 2 $$\nu^{9}$$ $$=$$ $$( -3715\beta_{9} - 1105\beta_{8} - 16354\beta_{6} + 16218\beta_{4} + 16218\beta_{3} + 23582\beta_1 ) / 2$$ (-3715*b9 - 1105*b8 - 16354*b6 + 16218*b4 + 16218*b3 + 23582*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$$.

 $$n$$ $$1093$$ $$1249$$ $$1457$$ $$2017$$ $$3823$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.455234i 0.812474i 1.21919i 2.54035i 3.49183i − 3.49183i − 2.54035i − 1.21919i − 0.812474i − 0.455234i
0 −1.00000 0 4.39335i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 2.46162i 0 1.00000i 0 1.00000 0
337.3 0 −1.00000 0 1.64044i 0 1.00000i 0 1.00000 0
337.4 0 −1.00000 0 0.787293i 0 1.00000i 0 1.00000 0
337.5 0 −1.00000 0 0.572766i 0 1.00000i 0 1.00000 0
337.6 0 −1.00000 0 0.572766i 0 1.00000i 0 1.00000 0
337.7 0 −1.00000 0 0.787293i 0 1.00000i 0 1.00000 0
337.8 0 −1.00000 0 1.64044i 0 1.00000i 0 1.00000 0
337.9 0 −1.00000 0 2.46162i 0 1.00000i 0 1.00000 0
337.10 0 −1.00000 0 4.39335i 0 1.00000i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.s 10
4.b odd 2 1 2184.2.h.g 10
13.b even 2 1 inner 4368.2.h.s 10
52.b odd 2 1 2184.2.h.g 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.h.g 10 4.b odd 2 1
2184.2.h.g 10 52.b odd 2 1
4368.2.h.s 10 1.a even 1 1 trivial
4368.2.h.s 10 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4368, [\chi])$$:

 $$T_{5}^{10} + 29T_{5}^{8} + 212T_{5}^{6} + 496T_{5}^{4} + 336T_{5}^{2} + 64$$ T5^10 + 29*T5^8 + 212*T5^6 + 496*T5^4 + 336*T5^2 + 64 $$T_{11}^{10} + 73T_{11}^{8} + 1576T_{11}^{6} + 11200T_{11}^{4} + 12944T_{11}^{2} + 4096$$ T11^10 + 73*T11^8 + 1576*T11^6 + 11200*T11^4 + 12944*T11^2 + 4096 $$T_{17}^{5} + 11T_{17}^{4} + 18T_{17}^{3} - 96T_{17}^{2} - 128T_{17} + 256$$ T17^5 + 11*T17^4 + 18*T17^3 - 96*T17^2 - 128*T17 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$(T + 1)^{10}$$
$5$ $$T^{10} + 29 T^{8} + 212 T^{6} + \cdots + 64$$
$7$ $$(T^{2} + 1)^{5}$$
$11$ $$T^{10} + 73 T^{8} + 1576 T^{6} + \cdots + 4096$$
$13$ $$T^{10} - 23 T^{8} + 64 T^{7} + \cdots + 371293$$
$17$ $$(T^{5} + 11 T^{4} + 18 T^{3} - 96 T^{2} + \cdots + 256)^{2}$$
$19$ $$T^{10} + 177 T^{8} + \cdots + 39337984$$
$23$ $$(T^{5} - 9 T^{4} - 8 T^{3} + 96 T^{2} + \cdots - 16)^{2}$$
$29$ $$(T^{5} + T^{4} - 30 T^{3} - 20 T^{2} + \cdots + 128)^{2}$$
$31$ $$T^{10}$$
$37$ $$T^{10} + 109 T^{8} + 3284 T^{6} + \cdots + 4096$$
$41$ $$T^{10} + 256 T^{8} + 19072 T^{6} + \cdots + 6553600$$
$43$ $$(T^{5} - T^{4} - 148 T^{3} + 160 T^{2} + \cdots + 2048)^{2}$$
$47$ $$T^{10} + 196 T^{8} + \cdots + 16777216$$
$53$ $$(T^{5} - 160 T^{3} + 304 T^{2} + \cdots - 9920)^{2}$$
$59$ $$T^{10} + 316 T^{8} + 29984 T^{6} + \cdots + 2408704$$
$61$ $$(T^{5} - 3 T^{4} - 110 T^{3} + 316 T^{2} + \cdots - 4192)^{2}$$
$67$ $$T^{10} + 420 T^{8} + 53632 T^{6} + \cdots + 262144$$
$71$ $$T^{10} + 316 T^{8} + \cdots + 203689984$$
$73$ $$T^{10} + 277 T^{8} + \cdots + 50922496$$
$79$ $$(T^{5} + 20 T^{4} - 8 T^{3} - 1168 T^{2} + \cdots + 5120)^{2}$$
$83$ $$T^{10} + 492 T^{8} + \cdots + 1774768384$$
$89$ $$T^{10} + 448 T^{8} + \cdots + 10240000$$
$97$ $$T^{10} + 328 T^{8} + \cdots + 125440000$$