# Properties

 Label 4368.2.h.r 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: $$x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36$$ x^10 + 20*x^8 + 138*x^6 + 364*x^4 + 249*x^2 + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - \beta_{4} q^{5} - \beta_{7} q^{7} + q^{9}+O(q^{10})$$ q - q^3 - b4 * q^5 - b7 * q^7 + q^9 $$q - q^{3} - \beta_{4} q^{5} - \beta_{7} q^{7} + q^{9} + ( - \beta_{9} + \beta_{3}) q^{11} + ( - \beta_{6} - \beta_{4} - \beta_{3}) q^{13} + \beta_{4} q^{15} + (\beta_{5} + \beta_1 + 1) q^{17} + (\beta_{8} + \beta_{7} + \beta_{6} + \beta_{3}) q^{19} + \beta_{7} q^{21} + (2 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + 1) q^{23} + \beta_{2} q^{25} - q^{27} + ( - \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{2} - 5) q^{29} + ( - \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3}) q^{31} + (\beta_{9} - \beta_{3}) q^{33} - \beta_{5} q^{35} + ( - \beta_{9} + \beta_{7} - \beta_{4}) q^{37} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{39} + ( - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3}) q^{41} + ( - \beta_{8} + \beta_{6} - \beta_{2} + 2 \beta_1 - 3) q^{43} - \beta_{4} q^{45} + (3 \beta_{9} + \beta_{8} + 5 \beta_{7} + \beta_{6} + 2 \beta_{4}) q^{47} - q^{49} + ( - \beta_{5} - \beta_1 - 1) q^{51} + ( - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{53} + (3 \beta_{8} - 3 \beta_{6} + \beta_{5} + 2 \beta_{2} - \beta_1 + 3) q^{55} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{3}) q^{57} + (2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \beta_{4} + 3 \beta_{3}) q^{59} + (\beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_1 + 3) q^{61} - \beta_{7} q^{63} + (\beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 4) q^{65} + ( - 2 \beta_{9} + 2 \beta_{8} + 6 \beta_{7} + 2 \beta_{6}) q^{67} + ( - 2 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} - \beta_{2} - 1) q^{69} + ( - 5 \beta_{7} + \beta_{4} + \beta_{3}) q^{71} + (3 \beta_{7} - 2 \beta_{4} + \beta_{3}) q^{73} - \beta_{2} q^{75} + (\beta_{2} + \beta_1) q^{77} + ( - \beta_{5} + \beta_{2} - 3 \beta_1) q^{79} + q^{81} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 4 \beta_{4}) q^{83} + ( - \beta_{9} - 2 \beta_{8} - 7 \beta_{7} - 2 \beta_{6} - 3 \beta_{4}) q^{85} + (\beta_{8} - \beta_{6} + 2 \beta_{5} - \beta_{2} + 5) q^{87} + (\beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - 4 \beta_{4}) q^{89} + ( - \beta_{8} - \beta_{5} - \beta_{2}) q^{91} + (\beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{93} + (\beta_{8} - \beta_{6} + 3 \beta_{2} + 2 \beta_1 - 3) q^{95} + (\beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + 3 \beta_{4} + 3 \beta_{3}) q^{97} + ( - \beta_{9} + \beta_{3}) q^{99}+O(q^{100})$$ q - q^3 - b4 * q^5 - b7 * q^7 + q^9 + (-b9 + b3) * q^11 + (-b6 - b4 - b3) * q^13 + b4 * q^15 + (b5 + b1 + 1) * q^17 + (b8 + b7 + b6 + b3) * q^19 + b7 * q^21 + (2*b8 - 2*b6 + 2*b5 + b2 + 1) * q^23 + b2 * q^25 - q^27 + (-b8 + b6 - 2*b5 + b2 - 5) * q^29 + (-b9 - b8 - 2*b7 - b6 - b4 + b3) * q^31 + (b9 - b3) * q^33 - b5 * q^35 + (-b9 + b7 - b4) * q^37 + (b6 + b4 + b3) * q^39 + (-2*b9 - b8 + b7 - b6 + b4 - b3) * q^41 + (-b8 + b6 - b2 + 2*b1 - 3) * q^43 - b4 * q^45 + (3*b9 + b8 + 5*b7 + b6 + 2*b4) * q^47 - q^49 + (-b5 - b1 - 1) * q^51 + (-b8 + b6 + b5 - b2 - b1) * q^53 + (3*b8 - 3*b6 + b5 + 2*b2 - b1 + 3) * q^55 + (-b8 - b7 - b6 - b3) * q^57 + (2*b8 + 3*b7 + 2*b6 + b4 + 3*b3) * q^59 + (b8 - b6 + b5 + 2*b2 + b1 + 3) * q^61 - b7 * q^63 + (b9 - b8 + 2*b7 + b6 + b5 - b3 - b2 - b1 - 4) * q^65 + (-2*b9 + 2*b8 + 6*b7 + 2*b6) * q^67 + (-2*b8 + 2*b6 - 2*b5 - b2 - 1) * q^69 + (-5*b7 + b4 + b3) * q^71 + (3*b7 - 2*b4 + b3) * q^73 - b2 * q^75 + (b2 + b1) * q^77 + (-b5 + b2 - 3*b1) * q^79 + q^81 + (-b9 - b8 + b7 - b6 + 4*b4) * q^83 + (-b9 - 2*b8 - 7*b7 - 2*b6 - 3*b4) * q^85 + (b8 - b6 + 2*b5 - b2 + 5) * q^87 + (b9 - 2*b8 - b7 - 2*b6 - 4*b4) * q^89 + (-b8 - b5 - b2) * q^91 + (b9 + b8 + 2*b7 + b6 + b4 - b3) * q^93 + (b8 - b6 + 3*b2 + 2*b1 - 3) * q^95 + (b9 + 4*b8 + 4*b7 + 4*b6 + 3*b4 + 3*b3) * q^97 + (-b9 + b3) * 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 $$10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100})$$ 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{2}$$ $$=$$ $$( \nu^{8} + 11\nu^{6} + 27\nu^{4} - 11\nu^{2} - 12 ) / 12$$ (v^8 + 11*v^6 + 27*v^4 - 11*v^2 - 12) / 12 $$\beta_{3}$$ $$=$$ $$( \nu^{9} + 17\nu^{7} + 87\nu^{5} + 139\nu^{3} + 84\nu ) / 36$$ (v^9 + 17*v^7 + 87*v^5 + 139*v^3 + 84*v) / 36 $$\beta_{4}$$ $$=$$ $$( \nu^{9} + 17\nu^{7} + 105\nu^{5} + 283\nu^{3} + 246\nu ) / 36$$ (v^9 + 17*v^7 + 105*v^5 + 283*v^3 + 246*v) / 36 $$\beta_{5}$$ $$=$$ $$( \nu^{8} + 11\nu^{6} + 15\nu^{4} - 95\nu^{2} - 48 ) / 12$$ (v^8 + 11*v^6 + 15*v^4 - 95*v^2 - 48) / 12 $$\beta_{6}$$ $$=$$ $$( \nu^{8} + 14\nu^{6} + 57\nu^{4} + 58\nu^{2} + 6\nu + 12 ) / 6$$ (v^8 + 14*v^6 + 57*v^4 + 58*v^2 + 6*v + 12) / 6 $$\beta_{7}$$ $$=$$ $$( -\nu^{9} - 17\nu^{7} - 96\nu^{5} - 193\nu^{3} - 75\nu ) / 18$$ (-v^9 - 17*v^7 - 96*v^5 - 193*v^3 - 75*v) / 18 $$\beta_{8}$$ $$=$$ $$( -\nu^{8} - 14\nu^{6} - 57\nu^{4} - 58\nu^{2} + 6\nu - 12 ) / 6$$ (-v^8 - 14*v^6 - 57*v^4 - 58*v^2 + 6*v - 12) / 6 $$\beta_{9}$$ $$=$$ $$( \nu^{9} + 26\nu^{7} + 213\nu^{5} + 598\nu^{3} + 264\nu ) / 18$$ (v^9 + 26*v^7 + 213*v^5 + 598*v^3 + 264*v) / 18
 $$\nu$$ $$=$$ $$( \beta_{8} + \beta_{6} ) / 2$$ (b8 + b6) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 - 4$$ b1 - 4 $$\nu^{3}$$ $$=$$ $$( -5\beta_{8} + 2\beta_{7} - 5\beta_{6} + 2\beta_{4} + 2\beta_{3} ) / 2$$ (-5*b8 + 2*b7 - 5*b6 + 2*b4 + 2*b3) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{5} + \beta_{2} - 7\beta _1 + 25$$ -b5 + b2 - 7*b1 + 25 $$\nu^{5}$$ $$=$$ $$( 31\beta_{8} - 16\beta_{7} + 31\beta_{6} - 12\beta_{4} - 20\beta_{3} ) / 2$$ (31*b8 - 16*b7 + 31*b6 - 12*b4 - 20*b3) / 2 $$\nu^{6}$$ $$=$$ $$-\beta_{8} + \beta_{6} + 10\beta_{5} - 14\beta_{2} + 47\beta _1 - 166$$ -b8 + b6 + 10*b5 - 14*b2 + 47*b1 - 166 $$\nu^{7}$$ $$=$$ $$( 4\beta_{9} - 199\beta_{8} + 122\beta_{7} - 199\beta_{6} + 66\beta_{4} + 170\beta_{3} ) / 2$$ (4*b9 - 199*b8 + 122*b7 - 199*b6 + 66*b4 + 170*b3) / 2 $$\nu^{8}$$ $$=$$ $$11\beta_{8} - 11\beta_{6} - 83\beta_{5} + 139\beta_{2} - 317\beta _1 + 1119$$ 11*b8 - 11*b6 - 83*b5 + 139*b2 - 317*b1 + 1119 $$\nu^{9}$$ $$=$$ $$( -68\beta_{9} + 1297\beta_{8} - 960\beta_{7} + 1297\beta_{6} - 356\beta_{4} - 1356\beta_{3} ) / 2$$ (-68*b9 + 1297*b8 - 960*b7 + 1297*b6 - 356*b4 - 1356*b3) / 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
 − 2.28392i − 2.56826i 0.444296i 0.854441i − 2.69449i 2.69449i − 0.854441i − 0.444296i 2.56826i 2.28392i
0 −1.00000 0 2.89917i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 2.71697i 0 1.00000i 0 1.00000 0
337.3 0 −1.00000 0 2.39548i 0 1.00000i 0 1.00000 0
337.4 0 −1.00000 0 2.11295i 0 1.00000i 0 1.00000 0
337.5 0 −1.00000 0 0.100328i 0 1.00000i 0 1.00000 0
337.6 0 −1.00000 0 0.100328i 0 1.00000i 0 1.00000 0
337.7 0 −1.00000 0 2.11295i 0 1.00000i 0 1.00000 0
337.8 0 −1.00000 0 2.39548i 0 1.00000i 0 1.00000 0
337.9 0 −1.00000 0 2.71697i 0 1.00000i 0 1.00000 0
337.10 0 −1.00000 0 2.89917i 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.r 10
4.b odd 2 1 2184.2.h.f 10
13.b even 2 1 inner 4368.2.h.r 10
52.b odd 2 1 2184.2.h.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.h.f 10 4.b odd 2 1
2184.2.h.f 10 52.b odd 2 1
4368.2.h.r 10 1.a even 1 1 trivial
4368.2.h.r 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 $$S_{2}^{\mathrm{new}}(4368, [\chi])$$:

 $$T_{5}^{10} + 26T_{5}^{8} + 249T_{5}^{6} + 1040T_{5}^{4} + 1600T_{5}^{2} + 16$$ T5^10 + 26*T5^8 + 249*T5^6 + 1040*T5^4 + 1600*T5^2 + 16 $$T_{11}^{10} + 73T_{11}^{8} + 1976T_{11}^{6} + 24192T_{11}^{4} + 128528T_{11}^{2} + 215296$$ T11^10 + 73*T11^8 + 1976*T11^6 + 24192*T11^4 + 128528*T11^2 + 215296 $$T_{17}^{5} - 5T_{17}^{4} - 26T_{17}^{3} + 104T_{17}^{2} + 96T_{17} - 384$$ T17^5 - 5*T17^4 - 26*T17^3 + 104*T17^2 + 96*T17 - 384

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$(T + 1)^{10}$$
$5$ $$T^{10} + 26 T^{8} + 249 T^{6} + \cdots + 16$$
$7$ $$(T^{2} + 1)^{5}$$
$11$ $$T^{10} + 73 T^{8} + 1976 T^{6} + \cdots + 215296$$
$13$ $$T^{10} + 4 T^{9} + 33 T^{8} + \cdots + 371293$$
$17$ $$(T^{5} - 5 T^{4} - 26 T^{3} + 104 T^{2} + \cdots - 384)^{2}$$
$19$ $$T^{10} + 74 T^{8} + 1401 T^{6} + \cdots + 1024$$
$23$ $$(T^{5} + 4 T^{4} - 87 T^{3} - 462 T^{2} + \cdots + 5272)^{2}$$
$29$ $$(T^{5} + 22 T^{4} + 103 T^{3} - 538 T^{2} + \cdots - 216)^{2}$$
$31$ $$T^{10} + 161 T^{8} + 8256 T^{6} + \cdots + 4194304$$
$37$ $$T^{10} + 73 T^{8} + 1248 T^{6} + \cdots + 16384$$
$41$ $$T^{10} + 324 T^{8} + \cdots + 707134464$$
$43$ $$(T^{5} + 10 T^{4} - 75 T^{3} - 480 T^{2} + \cdots - 2048)^{2}$$
$47$ $$T^{10} + 493 T^{8} + \cdots + 259081216$$
$53$ $$(T^{5} - 5 T^{4} - 88 T^{3} - 72 T^{2} + \cdots + 2032)^{2}$$
$59$ $$T^{10} + 360 T^{8} + \cdots + 131974144$$
$61$ $$(T^{5} - 9 T^{4} - 58 T^{3} + 372 T^{2} + \cdots - 1312)^{2}$$
$67$ $$T^{10} + 676 T^{8} + \cdots + 17314349056$$
$71$ $$T^{10} + 204 T^{8} + 12944 T^{6} + \cdots + 9216$$
$73$ $$T^{10} + 186 T^{8} + 12809 T^{6} + \cdots + 2027776$$
$79$ $$(T^{5} + T^{4} - 256 T^{3} - 680 T^{2} + \cdots + 35136)^{2}$$
$83$ $$T^{10} + 713 T^{8} + \cdots + 1802511936$$
$89$ $$T^{10} + 473 T^{8} + \cdots + 43243776$$
$97$ $$T^{10} + 757 T^{8} + \cdots + 1109689344$$