# Properties

 Label 363.2.f.g Level $363$ Weight $2$ Character orbit 363.f Analytic conductor $2.899$ Analytic rank $0$ Dimension $16$ CM discriminant -3 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.f (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: 16.0.6879707136000000000000.7 Defining polynomial: $$x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1$$ x^16 - 4*x^14 + 15*x^12 - 56*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$11^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{13} - \beta_{10} + \beta_{5} + \beta_1) q^{3} + 2 \beta_{4} q^{4} - \beta_{14} q^{7} + ( - 3 \beta_{12} + 3 \beta_{7} + 3 \beta_{4} - 3) q^{9}+O(q^{10})$$ q + (-b13 - b10 + b5 + b1) * q^3 + 2*b4 * q^4 - b14 * q^7 + (-3*b12 + 3*b7 + 3*b4 - 3) * q^9 $$q + ( - \beta_{13} - \beta_{10} + \beta_{5} + \beta_1) q^{3} + 2 \beta_{4} q^{4} - \beta_{14} q^{7} + ( - 3 \beta_{12} + 3 \beta_{7} + 3 \beta_{4} - 3) q^{9} + 2 \beta_1 q^{12} + \beta_{3} q^{13} - 4 \beta_{7} q^{16} + (\beta_{8} + \beta_{6}) q^{19} + (\beta_{11} + \beta_{9}) q^{21} + 5 \beta_{12} q^{25} + 3 \beta_{10} q^{27} + ( - 2 \beta_{14} + 2 \beta_{9} - 2 \beta_{6} + 2 \beta_{2}) q^{28} - \beta_{13} q^{31} - 6 \beta_{12} q^{36} + 3 \beta_{5} q^{37} + ( - \beta_{15} + 2 \beta_{14} - 2 \beta_{9} + 2 \beta_{6} - 2 \beta_{2}) q^{39} + ( - \beta_{11} - 2 \beta_{9}) q^{43} - 4 \beta_{5} q^{48} + ( - 5 \beta_{10} + 7 \beta_{7}) q^{49} - 2 \beta_{8} q^{52} + ( - 2 \beta_{3} + \beta_{2}) q^{57} + (2 \beta_{15} + \beta_{14} - \beta_{9} + \beta_{6} - \beta_{2}) q^{61} - 3 \beta_{6} q^{63} + (8 \beta_{12} - 8 \beta_{7} - 8 \beta_{4} + 8) q^{64} - 7 \beta_1 q^{67} + ( - \beta_{15} + 3 \beta_{14} - \beta_{11} - \beta_{8} - \beta_{3}) q^{73} + 5 \beta_{13} q^{75} + ( - 2 \beta_{11} + 2 \beta_{9}) q^{76} + ( - \beta_{3} - 3 \beta_{2}) q^{79} - 9 \beta_{4} q^{81} + (2 \beta_{15} + 2 \beta_{14} + 2 \beta_{11} + 2 \beta_{8} + 2 \beta_{3}) q^{84} + (9 \beta_{13} - \beta_{12} + 9 \beta_{10} - 9 \beta_{5} - 9 \beta_1) q^{91} + 3 \beta_{7} q^{93} + (5 \beta_{12} - 5 \beta_{7} - 5 \beta_{4} + 5) q^{97}+O(q^{100})$$ q + (-b13 - b10 + b5 + b1) * q^3 + 2*b4 * q^4 - b14 * q^7 + (-3*b12 + 3*b7 + 3*b4 - 3) * q^9 + 2*b1 * q^12 + b3 * q^13 - 4*b7 * q^16 + (b8 + b6) * q^19 + (b11 + b9) * q^21 + 5*b12 * q^25 + 3*b10 * q^27 + (-2*b14 + 2*b9 - 2*b6 + 2*b2) * q^28 - b13 * q^31 - 6*b12 * q^36 + 3*b5 * q^37 + (-b15 + 2*b14 - 2*b9 + 2*b6 - 2*b2) * q^39 + (-b11 - 2*b9) * q^43 - 4*b5 * q^48 + (-5*b10 + 7*b7) * q^49 - 2*b8 * q^52 + (-2*b3 + b2) * q^57 + (2*b15 + b14 - b9 + b6 - b2) * q^61 - 3*b6 * q^63 + (8*b12 - 8*b7 - 8*b4 + 8) * q^64 - 7*b1 * q^67 + (-b15 + 3*b14 - b11 - b8 - b3) * q^73 + 5*b13 * q^75 + (-2*b11 + 2*b9) * q^76 + (-b3 - 3*b2) * q^79 - 9*b4 * q^81 + (2*b15 + 2*b14 + 2*b11 + 2*b8 + 2*b3) * q^84 + (9*b13 - b12 + 9*b10 - 9*b5 - 9*b1) * q^91 + 3*b7 * q^93 + (5*b12 - 5*b7 - 5*b4 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} - 12 q^{9}+O(q^{10})$$ 16 * q + 8 * q^4 - 12 * q^9 $$16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100})$$ 16 * q + 8 * q^4 - 12 * q^9 - 16 * q^16 - 20 * q^25 + 24 * q^36 + 28 * q^49 + 32 * q^64 - 36 * q^81 + 4 * q^91 + 12 * q^93 + 20 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{10} + 362 ) / 209$$ (v^10 + 362) / 209 $$\beta_{2}$$ $$=$$ $$( -2\nu^{11} - 933\nu ) / 209$$ (-2*v^11 - 933*v) / 209 $$\beta_{3}$$ $$=$$ $$( 3\nu^{11} + 2549\nu ) / 209$$ (3*v^11 + 2549*v) / 209 $$\beta_{4}$$ $$=$$ $$( \nu^{12} + 780\nu^{2} ) / 209$$ (v^12 + 780*v^2) / 209 $$\beta_{5}$$ $$=$$ $$( -2\nu^{12} - 1351\nu^{2} ) / 209$$ (-2*v^12 - 1351*v^2) / 209 $$\beta_{6}$$ $$=$$ $$( -5\nu^{13} - 3482\nu^{3} ) / 209$$ (-5*v^13 - 3482*v^3) / 209 $$\beta_{7}$$ $$=$$ $$( -4\nu^{14} - 2911\nu^{4} ) / 209$$ (-4*v^14 - 2911*v^4) / 209 $$\beta_{8}$$ $$=$$ $$( -13\nu^{13} - 9513\nu^{3} ) / 209$$ (-13*v^13 - 9513*v^3) / 209 $$\beta_{9}$$ $$=$$ $$( -18\nu^{15} - 12995\nu^{5} ) / 209$$ (-18*v^15 - 12995*v^5) / 209 $$\beta_{10}$$ $$=$$ $$( -7\nu^{14} - 5042\nu^{4} ) / 209$$ (-7*v^14 - 5042*v^4) / 209 $$\beta_{11}$$ $$=$$ $$( 49\nu^{15} + 35503\nu^{5} ) / 209$$ (49*v^15 + 35503*v^5) / 209 $$\beta_{12}$$ $$=$$ $$( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209$$ (56*v^14 - 224*v^12 + 840*v^10 - 3135*v^8 + 11704*v^6 - 3136*v^4 + 840*v^2 - 224) / 209 $$\beta_{13}$$ $$=$$ $$( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209$$ (104*v^14 - 390*v^12 + 1456*v^10 - 5434*v^8 + 20273*v^6 - 390*v^4 + 104*v^2 - 26) / 209 $$\beta_{14}$$ $$=$$ $$( -268\nu^{15} + 1005\nu^{13} - 3752\nu^{11} + 14003\nu^{9} - 52250\nu^{7} + 1005\nu^{5} - 268\nu^{3} + 67\nu ) / 209$$ (-268*v^15 + 1005*v^13 - 3752*v^11 + 14003*v^9 - 52250*v^7 + 1005*v^5 - 268*v^3 + 67*v) / 209 $$\beta_{15}$$ $$=$$ $$( 683 \nu^{15} - 2732 \nu^{13} + 10245 \nu^{11} - 38247 \nu^{9} + 142747 \nu^{7} - 38248 \nu^{5} + 10245 \nu^{3} - 2732 \nu ) / 209$$ (683*v^15 - 2732*v^13 + 10245*v^11 - 38247*v^9 + 142747*v^7 - 38248*v^5 + 10245*v^3 - 2732*v) / 209
 $$\nu$$ $$=$$ $$( 2\beta_{3} + 3\beta_{2} ) / 11$$ (2*b3 + 3*b2) / 11 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{4}$$ b5 + 2*b4 $$\nu^{3}$$ $$=$$ $$( -5\beta_{8} + 13\beta_{6} ) / 11$$ (-5*b8 + 13*b6) / 11 $$\nu^{4}$$ $$=$$ $$4\beta_{10} - 7\beta_{7}$$ 4*b10 - 7*b7 $$\nu^{5}$$ $$=$$ $$( 18\beta_{11} + 49\beta_{9} ) / 11$$ (18*b11 + 49*b9) / 11 $$\nu^{6}$$ $$=$$ $$-15\beta_{13} + 26\beta_{12} - 26\beta_{7} - 26\beta_{4} + 26$$ -15*b13 + 26*b12 - 26*b7 - 26*b4 + 26 $$\nu^{7}$$ $$=$$ $$( 67\beta_{15} + 183\beta_{14} + 67\beta_{11} + 67\beta_{8} + 67\beta_{3} ) / 11$$ (67*b15 + 183*b14 + 67*b11 + 67*b8 + 67*b3) / 11 $$\nu^{8}$$ $$=$$ $$-56\beta_{13} + 97\beta_{12} - 56\beta_{10} + 56\beta_{5} + 56\beta_1$$ -56*b13 + 97*b12 - 56*b10 + 56*b5 + 56*b1 $$\nu^{9}$$ $$=$$ $$( 250\beta_{15} + 683\beta_{14} - 683\beta_{9} + 683\beta_{6} - 683\beta_{2} ) / 11$$ (250*b15 + 683*b14 - 683*b9 + 683*b6 - 683*b2) / 11 $$\nu^{10}$$ $$=$$ $$209\beta _1 - 362$$ 209*b1 - 362 $$\nu^{11}$$ $$=$$ $$( -933\beta_{3} - 2549\beta_{2} ) / 11$$ (-933*b3 - 2549*b2) / 11 $$\nu^{12}$$ $$=$$ $$-780\beta_{5} - 1351\beta_{4}$$ -780*b5 - 1351*b4 $$\nu^{13}$$ $$=$$ $$( 3482\beta_{8} - 9513\beta_{6} ) / 11$$ (3482*b8 - 9513*b6) / 11 $$\nu^{14}$$ $$=$$ $$-2911\beta_{10} + 5042\beta_{7}$$ -2911*b10 + 5042*b7 $$\nu^{15}$$ $$=$$ $$( -12995\beta_{11} - 35503\beta_{9} ) / 11$$ (-12995*b11 - 35503*b9) / 11

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −0.304260 + 0.418778i 0.304260 − 0.418778i 1.13551 − 1.56290i −1.13551 + 1.56290i −1.83730 + 0.596975i 1.83730 − 0.596975i 0.492303 − 0.159959i −0.492303 + 0.159959i −1.83730 − 0.596975i 1.83730 + 0.596975i 0.492303 + 0.159959i −0.492303 − 0.159959i −0.304260 − 0.418778i 0.304260 + 0.418778i 1.13551 + 1.56290i −1.13551 − 1.56290i
0 −0.535233 + 1.64728i −0.618034 1.90211i 0 0 −2.19769 + 0.714073i 0 −2.42705 1.76336i 0
161.2 0 −0.535233 + 1.64728i −0.618034 1.90211i 0 0 2.19769 0.714073i 0 −2.42705 1.76336i 0
161.3 0 0.535233 1.64728i −0.618034 1.90211i 0 0 −4.52729 + 1.47101i 0 −2.42705 1.76336i 0
161.4 0 0.535233 1.64728i −0.618034 1.90211i 0 0 4.52729 1.47101i 0 −2.42705 1.76336i 0
215.1 0 −1.40126 1.01807i 1.61803 1.17557i 0 0 −2.79802 3.85115i 0 0.927051 + 2.85317i 0
215.2 0 −1.40126 1.01807i 1.61803 1.17557i 0 0 2.79802 + 3.85115i 0 0.927051 + 2.85317i 0
215.3 0 1.40126 + 1.01807i 1.61803 1.17557i 0 0 −1.35825 1.86947i 0 0.927051 + 2.85317i 0
215.4 0 1.40126 + 1.01807i 1.61803 1.17557i 0 0 1.35825 + 1.86947i 0 0.927051 + 2.85317i 0
233.1 0 −1.40126 + 1.01807i 1.61803 + 1.17557i 0 0 −2.79802 + 3.85115i 0 0.927051 2.85317i 0
233.2 0 −1.40126 + 1.01807i 1.61803 + 1.17557i 0 0 2.79802 3.85115i 0 0.927051 2.85317i 0
233.3 0 1.40126 1.01807i 1.61803 + 1.17557i 0 0 −1.35825 + 1.86947i 0 0.927051 2.85317i 0
233.4 0 1.40126 1.01807i 1.61803 + 1.17557i 0 0 1.35825 1.86947i 0 0.927051 2.85317i 0
239.1 0 −0.535233 1.64728i −0.618034 + 1.90211i 0 0 −2.19769 0.714073i 0 −2.42705 + 1.76336i 0
239.2 0 −0.535233 1.64728i −0.618034 + 1.90211i 0 0 2.19769 + 0.714073i 0 −2.42705 + 1.76336i 0
239.3 0 0.535233 + 1.64728i −0.618034 + 1.90211i 0 0 −4.52729 1.47101i 0 −2.42705 + 1.76336i 0
239.4 0 0.535233 + 1.64728i −0.618034 + 1.90211i 0 0 4.52729 + 1.47101i 0 −2.42705 + 1.76336i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.g 16
3.b odd 2 1 CM 363.2.f.g 16
11.b odd 2 1 inner 363.2.f.g 16
11.c even 5 1 363.2.d.d 4
11.c even 5 3 inner 363.2.f.g 16
11.d odd 10 1 363.2.d.d 4
11.d odd 10 3 inner 363.2.f.g 16
33.d even 2 1 inner 363.2.f.g 16
33.f even 10 1 363.2.d.d 4
33.f even 10 3 inner 363.2.f.g 16
33.h odd 10 1 363.2.d.d 4
33.h odd 10 3 inner 363.2.f.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.d.d 4 11.c even 5 1
363.2.d.d 4 11.d odd 10 1
363.2.d.d 4 33.f even 10 1
363.2.d.d 4 33.h odd 10 1
363.2.f.g 16 1.a even 1 1 trivial
363.2.f.g 16 3.b odd 2 1 CM
363.2.f.g 16 11.b odd 2 1 inner
363.2.f.g 16 11.c even 5 3 inner
363.2.f.g 16 11.d odd 10 3 inner
363.2.f.g 16 33.d even 2 1 inner
363.2.f.g 16 33.f even 10 3 inner
363.2.f.g 16 33.h odd 10 3 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7}^{16} - 28 T_{7}^{14} + 663 T_{7}^{12} - 15176 T_{7}^{10} + 344705 T_{7}^{8} - 1836296 T_{7}^{6} + 9706983 T_{7}^{4} - 49603708 T_{7}^{2} + 214358881$$ T7^16 - 28*T7^14 + 663*T7^12 - 15176*T7^10 + 344705*T7^8 - 1836296*T7^6 + 9706983*T7^4 - 49603708*T7^2 + 214358881

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$(T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2}$$
$5$ $$T^{16}$$
$7$ $$T^{16} - 28 T^{14} + \cdots + 214358881$$
$11$ $$T^{16}$$
$13$ $$T^{16} - 52 T^{14} + \cdots + 54875873536$$
$17$ $$T^{16}$$
$19$ $$T^{16} - 76 T^{14} + \cdots + 214358881$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$(T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2}$$
$37$ $$(T^{8} + 27 T^{6} + 729 T^{4} + \cdots + 531441)^{2}$$
$41$ $$T^{16}$$
$43$ $$(T^{4} + 172 T^{2} + 484)^{4}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$T^{16}$$
$61$ $$T^{16} - 244 T^{14} + \cdots + 45\!\cdots\!61$$
$67$ $$(T^{2} - 147)^{8}$$
$71$ $$T^{16}$$
$73$ $$T^{16} - 292 T^{14} + \cdots + 17\!\cdots\!01$$
$79$ $$T^{16} - 316 T^{14} + \cdots + 214358881$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$(T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625)^{4}$$