# Properties

 Label 4368.2.h.t 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: $$x^{10} - 4x^{9} + 8x^{8} + 18x^{7} + 28x^{6} - 48x^{5} + 130x^{4} + 316x^{3} + 324x^{2} + 144x + 32$$ x^10 - 4*x^9 + 8*x^8 + 18*x^7 + 28*x^6 - 48*x^5 + 130*x^4 + 316*x^3 + 324*x^2 + 144*x + 32 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + (\beta_{5} + \beta_1) q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (b5 + b1) * q^5 + b1 * q^7 + q^9 $$q + q^{3} + (\beta_{5} + \beta_1) q^{5} + \beta_1 q^{7} + q^{9} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{11} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} + \beta_1) q^{15} - \beta_{8} q^{17} + ( - \beta_{9} + \beta_{6} - \beta_{4} - 2 \beta_1) q^{19} + \beta_1 q^{21} + ( - \beta_{8} - \beta_{2}) q^{23} + (\beta_{8} + \beta_{2} - 1) q^{25} + q^{27} + (\beta_{9} - \beta_{8} - \beta_{4} - \beta_{2}) q^{29} + ( - \beta_{9} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4}) q^{31} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{33} + ( - \beta_{3} - 1) q^{35} + (\beta_{7} + 2 \beta_{6}) q^{37} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{39} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{41} + ( - \beta_{9} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 4) q^{43} + (\beta_{5} + \beta_1) q^{45} + ( - \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_1) q^{47} - q^{49} - \beta_{8} q^{51} + (\beta_{9} + \beta_{8} - \beta_{4} - \beta_{2}) q^{53} + (\beta_{9} + \beta_{8} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 6) q^{55} + ( - \beta_{9} + \beta_{6} - \beta_{4} - 2 \beta_1) q^{57} + (\beta_{6} + \beta_{5} + \beta_1) q^{59} + (\beta_{9} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{61} + \beta_1 q^{63} + (\beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 3 \beta_1) q^{65} + ( - 2 \beta_{9} + \beta_{7} - 2 \beta_{4}) q^{67} + ( - \beta_{8} - \beta_{2}) q^{69} + (\beta_{6} - \beta_{5} - 7 \beta_1) q^{71} + ( - 2 \beta_{7} + 3 \beta_{6} - 2 \beta_1) q^{73} + (\beta_{8} + \beta_{2} - 1) q^{75} + ( - \beta_{3} + \beta_{2} + 1) q^{77} + (\beta_{8} + 2 \beta_{3} + 3 \beta_{2} + 4) q^{79} + q^{81} + ( - \beta_{9} - \beta_{5} - \beta_{4} + 5 \beta_1) q^{83} + (2 \beta_{6} + 4 \beta_{5}) q^{85} + (\beta_{9} - \beta_{8} - \beta_{4} - \beta_{2}) q^{87} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \beta_1) q^{89} + (\beta_{9} - \beta_{6} - \beta_{5}) q^{91} + ( - \beta_{9} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4}) q^{93} + (\beta_{9} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 4) q^{95} + (2 \beta_{9} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4}) q^{97} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{99}+O(q^{100})$$ q + q^3 + (b5 + b1) * q^5 + b1 * q^7 + q^9 + (-b6 + b5 - b1) * q^11 + (-b4 - b3 - b2) * q^13 + (b5 + b1) * q^15 - b8 * q^17 + (-b9 + b6 - b4 - 2*b1) * q^19 + b1 * q^21 + (-b8 - b2) * q^23 + (b8 + b2 - 1) * q^25 + q^27 + (b9 - b8 - b4 - b2) * q^29 + (-b9 + 3*b6 + 2*b5 - b4) * q^31 + (-b6 + b5 - b1) * q^33 + (-b3 - 1) * q^35 + (b7 + 2*b6) * q^37 + (-b4 - b3 - b2) * q^39 + (-b9 + b7 - b6 - b5 - b4 + b1) * q^41 + (-b9 + b4 + 2*b3 + b2 + 4) * q^43 + (b5 + b1) * q^45 + (-b9 - b7 - b5 - b4 + b1) * q^47 - q^49 - b8 * q^51 + (b9 + b8 - b4 - b2) * q^53 + (b9 + b8 - b4 + 2*b3 + 2*b2 - 6) * q^55 + (-b9 + b6 - b4 - 2*b1) * q^57 + (b6 + b5 + b1) * q^59 + (b9 - b4 - 2*b3 - 2*b2) * q^61 + b1 * q^63 + (b9 - b7 + b6 + b5 - b4 + b2 - 3*b1) * q^65 + (-2*b9 + b7 - 2*b4) * q^67 + (-b8 - b2) * q^69 + (b6 - b5 - 7*b1) * q^71 + (-2*b7 + 3*b6 - 2*b1) * q^73 + (b8 + b2 - 1) * q^75 + (-b3 + b2 + 1) * q^77 + (b8 + 2*b3 + 3*b2 + 4) * q^79 + q^81 + (-b9 - b5 - b4 + 5*b1) * q^83 + (2*b6 + 4*b5) * q^85 + (b9 - b8 - b4 - b2) * q^87 + (-b7 + 2*b6 - 3*b5 + b1) * q^89 + (b9 - b6 - b5) * q^91 + (-b9 + 3*b6 + 2*b5 - b4) * q^93 + (b9 - b4 + 2*b3 + b2 + 4) * q^95 + (2*b9 - b6 + 2*b5 + 2*b4) * q^97 + (-b6 + b5 - 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 $$10 q + 10 q^{3} + 10 q^{9} + 2 q^{13} - 2 q^{23} - 8 q^{25} + 10 q^{27} - 2 q^{29} - 6 q^{35} + 2 q^{39} + 34 q^{43} - 10 q^{49} - 2 q^{53} - 64 q^{55} + 4 q^{61} + 2 q^{65} - 2 q^{69} - 8 q^{75} + 16 q^{77} + 38 q^{79} + 10 q^{81} - 2 q^{87} + 34 q^{95}+O(q^{100})$$ 10 * q + 10 * q^3 + 10 * q^9 + 2 * q^13 - 2 * q^23 - 8 * q^25 + 10 * q^27 - 2 * q^29 - 6 * q^35 + 2 * q^39 + 34 * q^43 - 10 * q^49 - 2 * q^53 - 64 * q^55 + 4 * q^61 + 2 * q^65 - 2 * q^69 - 8 * q^75 + 16 * q^77 + 38 * q^79 + 10 * q^81 - 2 * q^87 + 34 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4x^{9} + 8x^{8} + 18x^{7} + 28x^{6} - 48x^{5} + 130x^{4} + 316x^{3} + 324x^{2} + 144x + 32$$ :

 $$\beta_{1}$$ $$=$$ $$( - 2335639 \nu^{9} + 10122488 \nu^{8} - 22385964 \nu^{7} - 32822558 \nu^{6} - 58610700 \nu^{5} + 129322072 \nu^{4} - 349494182 \nu^{3} + \cdots - 176351608 ) / 105963352$$ (-2335639*v^9 + 10122488*v^8 - 22385964*v^7 - 32822558*v^6 - 58610700*v^5 + 129322072*v^4 - 349494182*v^3 - 572373740*v^2 - 622168788*v - 176351608) / 105963352 $$\beta_{2}$$ $$=$$ $$( 2199633 \nu^{9} - 11134171 \nu^{8} + 27719552 \nu^{7} + 17207430 \nu^{6} + 28767166 \nu^{5} - 164193084 \nu^{4} + 415274362 \nu^{3} + \cdots - 199458284 ) / 52981676$$ (2199633*v^9 - 11134171*v^8 + 27719552*v^7 + 17207430*v^6 + 28767166*v^5 - 164193084*v^4 + 415274362*v^3 + 345589846*v^2 + 140307352*v - 199458284) / 52981676 $$\beta_{3}$$ $$=$$ $$( - 2589599 \nu^{9} + 12984597 \nu^{8} - 32329024 \nu^{7} - 20601026 \nu^{6} - 37372866 \nu^{5} + 187123640 \nu^{4} - 498118454 \nu^{3} + \cdots + 109106384 ) / 52981676$$ (-2589599*v^9 + 12984597*v^8 - 32329024*v^7 - 20601026*v^6 - 37372866*v^5 + 187123640*v^4 - 498118454*v^3 - 412878970*v^2 - 167315880*v + 109106384) / 52981676 $$\beta_{4}$$ $$=$$ $$( 154047 \nu^{9} - 438600 \nu^{8} + 335220 \nu^{7} + 5014126 \nu^{6} + 5698812 \nu^{5} - 5110072 \nu^{4} + 7742894 \nu^{3} + 82105292 \nu^{2} + \cdots + 33487184 ) / 2464264$$ (154047*v^9 - 438600*v^8 + 335220*v^7 + 5014126*v^6 + 5698812*v^5 - 5110072*v^4 + 7742894*v^3 + 82105292*v^2 + 77746676*v + 33487184) / 2464264 $$\beta_{5}$$ $$=$$ $$( - 114626 \nu^{9} + 478857 \nu^{8} - 988670 \nu^{7} - 1948576 \nu^{6} - 2736774 \nu^{5} + 6166732 \nu^{4} - 15561424 \nu^{3} - 35174798 \nu^{2} + \cdots - 8054672 ) / 1232132$$ (-114626*v^9 + 478857*v^8 - 988670*v^7 - 1948576*v^6 - 2736774*v^5 + 6166732*v^4 - 15561424*v^3 - 35174798*v^2 - 28368376*v - 8054672) / 1232132 $$\beta_{6}$$ $$=$$ $$( 12973407 \nu^{9} - 55005042 \nu^{8} + 116630528 \nu^{7} + 207187286 \nu^{6} + 311184664 \nu^{5} - 705522136 \nu^{4} + 1853464830 \nu^{3} + \cdots + 943793848 ) / 105963352$$ (12973407*v^9 - 55005042*v^8 + 116630528*v^7 + 207187286*v^6 + 311184664*v^5 - 705522136*v^4 + 1853464830*v^3 + 3731984616*v^2 + 3327792884*v + 943793848) / 105963352 $$\beta_{7}$$ $$=$$ $$( - 7709981 \nu^{9} + 30449668 \nu^{8} - 55204652 \nu^{7} - 167118734 \nu^{6} - 159470092 \nu^{5} + 395900472 \nu^{4} - 936616882 \nu^{3} + \cdots - 494347560 ) / 52981676$$ (-7709981*v^9 + 30449668*v^8 - 55204652*v^7 - 167118734*v^6 - 159470092*v^5 + 395900472*v^4 - 936616882*v^3 - 2794949292*v^2 - 1739154716*v - 494347560) / 52981676 $$\beta_{8}$$ $$=$$ $$( 2071882 \nu^{9} - 10424657 \nu^{8} + 26161144 \nu^{7} + 16150117 \nu^{6} + 26504200 \nu^{5} - 134359104 \nu^{4} + 389598134 \nu^{3} + \cdots + 2304942 ) / 13245419$$ (2071882*v^9 - 10424657*v^8 + 26161144*v^7 + 16150117*v^6 + 26504200*v^5 - 134359104*v^4 + 389598134*v^3 + 324501498*v^2 + 131798056*v + 2304942) / 13245419 $$\beta_{9}$$ $$=$$ $$( 25921189 \nu^{9} - 115617736 \nu^{8} + 256644668 \nu^{7} + 367636130 \nu^{6} + 508354068 \nu^{5} - 1514922200 \nu^{4} + 4004866410 \nu^{3} + \cdots + 810078176 ) / 105963352$$ (25921189*v^9 - 115617736*v^8 + 256644668*v^7 + 367636130*v^6 + 508354068*v^5 - 1514922200*v^4 + 4004866410*v^3 + 6581131156*v^2 + 4580665148*v + 810078176) / 105963352
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2$$ (b6 + b5 + b3 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{9} + \beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{4} + 10\beta_1 ) / 2$$ (b9 + b7 + 2*b6 + 2*b5 + b4 + 10*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{8} + \beta_{7} + 3\beta_{6} + 5\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} + 10\beta _1 - 10$$ b8 + b7 + 3*b6 + 5*b5 + 3*b4 - 5*b3 - 3*b2 + 10*b1 - 10 $$\nu^{4}$$ $$=$$ $$-10\beta_{9} + 8\beta_{8} + 10\beta_{4} - 21\beta_{3} - 11\beta_{2} - 59$$ -10*b9 + 8*b8 + 10*b4 - 21*b3 - 11*b2 - 59 $$\nu^{5}$$ $$=$$ $$( - 114 \beta_{9} + 41 \beta_{8} - 41 \beta_{7} - 60 \beta_{6} - 132 \beta_{5} - 132 \beta_{3} - 60 \beta_{2} - 300 \beta _1 - 300 ) / 2$$ (-114*b9 + 41*b8 - 41*b7 - 60*b6 - 132*b5 - 132*b3 - 60*b2 - 300*b1 - 300) / 2 $$\nu^{6}$$ $$=$$ $$-164\beta_{9} - 123\beta_{7} - 144\beta_{6} - 340\beta_{5} - 164\beta_{4} - 838\beta_1$$ -164*b9 - 123*b7 - 144*b6 - 340*b5 - 164*b4 - 838*b1 $$\nu^{7}$$ $$=$$ $$- 334 \beta_{8} - 334 \beta_{7} - 393 \beta_{6} - 967 \beta_{5} - 914 \beta_{4} + 967 \beta_{3} + 393 \beta_{2} - 2263 \beta _1 + 2263$$ -334*b8 - 334*b7 - 393*b6 - 967*b5 - 914*b4 + 967*b3 + 393*b2 - 2263*b1 + 2263 $$\nu^{8}$$ $$=$$ $$2549\beta_{9} - 1881\beta_{8} - 2549\beta_{4} + 5246\beta_{3} + 2082\beta_{2} + 12514$$ 2549*b9 - 1881*b8 - 2549*b4 + 5246*b3 + 2082*b2 + 12514 $$\nu^{9}$$ $$=$$ $$14106 \beta_{9} - 5172 \beta_{8} + 5172 \beta_{7} + 5716 \beta_{6} + 14576 \beta_{5} + 14576 \beta_{3} + 5716 \beta_{2} + 34330 \beta _1 + 34330$$ 14106*b9 - 5172*b8 + 5172*b7 + 5716*b6 + 14576*b5 + 14576*b3 + 5716*b2 + 34330*b1 + 34330

## 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
 −1.35191 − 1.35191i −0.340491 + 0.340491i 2.75844 − 2.75844i −0.535829 + 0.535829i 1.46979 + 1.46979i 1.46979 − 1.46979i −0.535829 − 0.535829i 2.75844 + 2.75844i −0.340491 − 0.340491i −1.35191 + 1.35191i
0 1.00000 0 3.22444i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 3.19289i 0 1.00000i 0 1.00000 0
337.3 0 1.00000 0 2.79184i 0 1.00000i 0 1.00000 0
337.4 0 1.00000 0 0.660872i 0 1.00000i 0 1.00000 0
337.5 0 1.00000 0 0.421158i 0 1.00000i 0 1.00000 0
337.6 0 1.00000 0 0.421158i 0 1.00000i 0 1.00000 0
337.7 0 1.00000 0 0.660872i 0 1.00000i 0 1.00000 0
337.8 0 1.00000 0 2.79184i 0 1.00000i 0 1.00000 0
337.9 0 1.00000 0 3.19289i 0 1.00000i 0 1.00000 0
337.10 0 1.00000 0 3.22444i 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.t 10
4.b odd 2 1 2184.2.h.e 10
13.b even 2 1 inner 4368.2.h.t 10
52.b odd 2 1 2184.2.h.e 10

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

 $$T_{5}^{10} + 29T_{5}^{8} + 284T_{5}^{6} + 992T_{5}^{4} + 528T_{5}^{2} + 64$$ T5^10 + 29*T5^8 + 284*T5^6 + 992*T5^4 + 528*T5^2 + 64 $$T_{11}^{10} + 96T_{11}^{8} + 3168T_{11}^{6} + 39312T_{11}^{4} + 111872T_{11}^{2} + 16384$$ T11^10 + 96*T11^8 + 3168*T11^6 + 39312*T11^4 + 111872*T11^2 + 16384 $$T_{17}^{5} - 52T_{17}^{3} + 64T_{17}^{2} + 384T_{17} - 128$$ T17^5 - 52*T17^3 + 64*T17^2 + 384*T17 - 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$(T - 1)^{10}$$
$5$ $$T^{10} + 29 T^{8} + 284 T^{6} + \cdots + 64$$
$7$ $$(T^{2} + 1)^{5}$$
$11$ $$T^{10} + 96 T^{8} + 3168 T^{6} + \cdots + 16384$$
$13$ $$T^{10} - 2 T^{9} - 11 T^{8} + \cdots + 371293$$
$17$ $$(T^{5} - 52 T^{3} + 64 T^{2} + 384 T - 128)^{2}$$
$19$ $$T^{10} + 97 T^{8} + 3040 T^{6} + \cdots + 16384$$
$23$ $$(T^{5} + T^{4} - 52 T^{3} + 16 T^{2} + \cdots - 1072)^{2}$$
$29$ $$(T^{5} + T^{4} - 90 T^{3} - 108 T^{2} + \cdots + 2144)^{2}$$
$31$ $$T^{10} + 241 T^{8} + 20160 T^{6} + \cdots + 4194304$$
$37$ $$T^{10} + 324 T^{8} + \cdots + 409010176$$
$41$ $$T^{10} + 268 T^{8} + 25520 T^{6} + \cdots + 5456896$$
$43$ $$(T^{5} - 17 T^{4} + 44 T^{3} + 320 T^{2} + \cdots - 1024)^{2}$$
$47$ $$T^{10} + 269 T^{8} + 22428 T^{6} + \cdots + 2027776$$
$53$ $$(T^{5} + T^{4} - 94 T^{3} - 532 T^{2} + \cdots - 256)^{2}$$
$59$ $$T^{10} + 48 T^{8} + 624 T^{6} + \cdots + 1024$$
$61$ $$(T^{5} - 2 T^{4} - 76 T^{3} + 40 T^{2} + \cdots + 2176)^{2}$$
$67$ $$T^{10} + 456 T^{8} + \cdots + 635846656$$
$71$ $$T^{10} + 288 T^{8} + \cdots + 11075584$$
$73$ $$T^{10} + 541 T^{8} + \cdots + 1628283904$$
$79$ $$(T^{5} - 19 T^{4} - 28 T^{3} + 1608 T^{2} + \cdots - 23168)^{2}$$
$83$ $$T^{10} + 281 T^{8} + \cdots + 66455104$$
$89$ $$T^{10} + 537 T^{8} + \cdots + 1514143744$$
$97$ $$T^{10} + 501 T^{8} + \cdots + 2252071936$$