# Properties

 Label 224.3.h.d Level $224$ Weight $3$ Character orbit 224.h Analytic conductor $6.104$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.976966189056.51 Defining polynomial: $$x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526$$ x^8 - 10*x^6 + 123*x^4 - 300*x^3 + 86*x^2 + 300*x + 526 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7} + (2 \beta_1 + 1) q^{9}+O(q^{10})$$ q - b3 * q^3 + (b3 - b2) * q^5 + (-b4 + b1 + 2) * q^7 + (2*b1 + 1) * q^9 $$q - \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7} + (2 \beta_1 + 1) q^{9} + ( - \beta_{6} - \beta_{5}) q^{11} + ( - \beta_{3} - 3 \beta_{2}) q^{13} + (\beta_1 - 8) q^{15} + (\beta_{7} - \beta_{4} + \beta_1) q^{17} + ( - 3 \beta_{3} - 4 \beta_{2}) q^{19} + ( - 2 \beta_{6} - 3 \beta_{5} - 3 \beta_{3} - \beta_{2}) q^{21} + ( - 6 \beta_1 + 4) q^{23} + ( - 6 \beta_1 - 3) q^{25} + (4 \beta_{3} - 4 \beta_{2}) q^{27} + ( - 2 \beta_{6} + 4 \beta_{5}) q^{29} + (2 \beta_{7} + \beta_1) q^{31} + ( - \beta_{7} - 3 \beta_{4} + \beta_1) q^{33} + ( - \beta_{6} + 5 \beta_{5} + \beta_{3}) q^{35} + (2 \beta_{6} - 2 \beta_{5}) q^{37} + (11 \beta_1 + 16) q^{39} + (4 \beta_{4} - 2 \beta_1) q^{41} + ( - 5 \beta_{6} - 9 \beta_{5}) q^{43} + ( - 3 \beta_{3} + 7 \beta_{2}) q^{45} + ( - 2 \beta_{7} - \beta_1) q^{47} + ( - \beta_{7} - 3 \beta_{4} + 5 \beta_1 - 35) q^{49} + (10 \beta_{6} + 2 \beta_{5}) q^{51} + 6 \beta_{6} q^{53} + (2 \beta_{4} - \beta_1) q^{55} + (18 \beta_1 + 38) q^{57} + (\beta_{3} + 12 \beta_{2}) q^{59} + (\beta_{3} - \beta_{2}) q^{61} + ( - 2 \beta_{7} + \beta_{4} + 3 \beta_1 + 14) q^{63} + ( - 14 \beta_1 + 34) q^{65} + (5 \beta_{6} + 9 \beta_{5}) q^{67} + (8 \beta_{3} + 12 \beta_{2}) q^{69} + ( - 13 \beta_1 - 68) q^{71} + (\beta_{7} + 11 \beta_{4} - 5 \beta_1) q^{73} + (15 \beta_{3} + 12 \beta_{2}) q^{75} + ( - 4 \beta_{6} - 3 \beta_{5} + 14 \beta_{3} + 2 \beta_{2}) q^{77} + ( - 9 \beta_1 + 4) q^{79} + ( - 14 \beta_1 - 41) q^{81} + ( - 17 \beta_{3} + 24 \beta_{2}) q^{83} + ( - 8 \beta_{6} + 14 \beta_{5}) q^{85} + ( - 2 \beta_{7} + 6 \beta_{4} - 4 \beta_1) q^{87} + ( - \beta_{7} + 13 \beta_{4} - 7 \beta_1) q^{89} + ( - 11 \beta_{6} + 3 \beta_{5} - 9 \beta_{3} - 4 \beta_{2}) q^{91} + (24 \beta_{6} + 10 \beta_{5}) q^{93} + ( - 17 \beta_1 + 32) q^{95} + (\beta_{7} - 17 \beta_{4} + 9 \beta_1) q^{97} + ( - 9 \beta_{6} - 5 \beta_{5}) q^{99}+O(q^{100})$$ q - b3 * q^3 + (b3 - b2) * q^5 + (-b4 + b1 + 2) * q^7 + (2*b1 + 1) * q^9 + (-b6 - b5) * q^11 + (-b3 - 3*b2) * q^13 + (b1 - 8) * q^15 + (b7 - b4 + b1) * q^17 + (-3*b3 - 4*b2) * q^19 + (-2*b6 - 3*b5 - 3*b3 - b2) * q^21 + (-6*b1 + 4) * q^23 + (-6*b1 - 3) * q^25 + (4*b3 - 4*b2) * q^27 + (-2*b6 + 4*b5) * q^29 + (2*b7 + b1) * q^31 + (-b7 - 3*b4 + b1) * q^33 + (-b6 + 5*b5 + b3) * q^35 + (2*b6 - 2*b5) * q^37 + (11*b1 + 16) * q^39 + (4*b4 - 2*b1) * q^41 + (-5*b6 - 9*b5) * q^43 + (-3*b3 + 7*b2) * q^45 + (-2*b7 - b1) * q^47 + (-b7 - 3*b4 + 5*b1 - 35) * q^49 + (10*b6 + 2*b5) * q^51 + 6*b6 * q^53 + (2*b4 - b1) * q^55 + (18*b1 + 38) * q^57 + (b3 + 12*b2) * q^59 + (b3 - b2) * q^61 + (-2*b7 + b4 + 3*b1 + 14) * q^63 + (-14*b1 + 34) * q^65 + (5*b6 + 9*b5) * q^67 + (8*b3 + 12*b2) * q^69 + (-13*b1 - 68) * q^71 + (b7 + 11*b4 - 5*b1) * q^73 + (15*b3 + 12*b2) * q^75 + (-4*b6 - 3*b5 + 14*b3 + 2*b2) * q^77 + (-9*b1 + 4) * q^79 + (-14*b1 - 41) * q^81 + (-17*b3 + 24*b2) * q^83 + (-8*b6 + 14*b5) * q^85 + (-2*b7 + 6*b4 - 4*b1) * q^87 + (-b7 + 13*b4 - 7*b1) * q^89 + (-11*b6 + 3*b5 - 9*b3 - 4*b2) * q^91 + (24*b6 + 10*b5) * q^93 + (-17*b1 + 32) * q^95 + (b7 - 17*b4 + 9*b1) * q^97 + (-9*b6 - 5*b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{7} + 8 q^{9}+O(q^{10})$$ 8 * q + 16 * q^7 + 8 * q^9 $$8 q + 16 q^{7} + 8 q^{9} - 64 q^{15} + 32 q^{23} - 24 q^{25} + 128 q^{39} - 280 q^{49} + 304 q^{57} + 112 q^{63} + 272 q^{65} - 544 q^{71} + 32 q^{79} - 328 q^{81} + 256 q^{95}+O(q^{100})$$ 8 * q + 16 * q^7 + 8 * q^9 - 64 * q^15 + 32 * q^23 - 24 * q^25 + 128 * q^39 - 280 * q^49 + 304 * q^57 + 112 * q^63 + 272 * q^65 - 544 * q^71 + 32 * q^79 - 328 * q^81 + 256 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526$$ :

 $$\beta_{1}$$ $$=$$ $$( - 868 \nu^{7} - 9000 \nu^{6} - 26304 \nu^{5} + 91850 \nu^{4} + 320592 \nu^{3} - 425950 \nu^{2} - 816140 \nu + 4767050 ) / 1582309$$ (-868*v^7 - 9000*v^6 - 26304*v^5 + 91850*v^4 + 320592*v^3 - 425950*v^2 - 816140*v + 4767050) / 1582309 $$\beta_{2}$$ $$=$$ $$( 2679764 \nu^{7} + 208186 \nu^{6} - 31814108 \nu^{5} - 10053780 \nu^{4} + 411714584 \nu^{3} - 624881484 \nu^{2} - 1121009602 \nu + 982876622 ) / 685139797$$ (2679764*v^7 + 208186*v^6 - 31814108*v^5 - 10053780*v^4 + 411714584*v^3 - 624881484*v^2 - 1121009602*v + 982876622) / 685139797 $$\beta_{3}$$ $$=$$ $$( 3872412 \nu^{7} + 6558661 \nu^{6} - 34456727 \nu^{5} - 55216902 \nu^{4} + 394683447 \nu^{3} - 633800883 \nu^{2} - 1116611684 \nu + 1714881404 ) / 685139797$$ (3872412*v^7 + 6558661*v^6 - 34456727*v^5 - 55216902*v^4 + 394683447*v^3 - 633800883*v^2 - 1116611684*v + 1714881404) / 685139797 $$\beta_{4}$$ $$=$$ $$( - 12992 \nu^{7} + 18417 \nu^{6} + 167753 \nu^{5} - 3353 \nu^{4} - 1377571 \nu^{3} + 4853780 \nu^{2} - 5682368 \nu + 556407 ) / 1582309$$ (-12992*v^7 + 18417*v^6 + 167753*v^5 - 3353*v^4 - 1377571*v^3 + 4853780*v^2 - 5682368*v + 556407) / 1582309 $$\beta_{5}$$ $$=$$ $$( 5735372 \nu^{7} + 4313372 \nu^{6} - 52238584 \nu^{5} - 59878610 \nu^{4} + 684612832 \nu^{3} - 1065326618 \nu^{2} + 851928604 \nu - 98379406 ) / 685139797$$ (5735372*v^7 + 4313372*v^6 - 52238584*v^5 - 59878610*v^4 + 684612832*v^3 - 1065326618*v^2 + 851928604*v - 98379406) / 685139797 $$\beta_{6}$$ $$=$$ $$( 8071492 \nu^{7} - 164500 \nu^{6} - 46895176 \nu^{5} + 109451638 \nu^{4} + 845538352 \nu^{3} - 2992899254 \nu^{2} + 2981272772 \nu + 1079942072 ) / 685139797$$ (8071492*v^7 - 164500*v^6 - 46895176*v^5 + 109451638*v^4 + 845538352*v^3 - 2992899254*v^2 + 2981272772*v + 1079942072) / 685139797 $$\beta_{7}$$ $$=$$ $$( - 19240 \nu^{7} + 143219 \nu^{6} + 372167 \nu^{5} - 1048471 \nu^{4} - 3401197 \nu^{3} + 23225460 \nu^{2} - 25469724 \nu - 15107251 ) / 1582309$$ (-19240*v^7 + 143219*v^6 + 372167*v^5 - 1048471*v^4 - 3401197*v^3 + 23225460*v^2 - 25469724*v - 15107251) / 1582309
 $$\nu$$ $$=$$ $$( \beta_{5} - 2\beta_{2} + \beta_1 ) / 4$$ (b5 - 2*b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - 4\beta_{3} + 2\beta_{2} + 2\beta _1 + 10 ) / 4$$ (b7 + b6 - b5 - 3*b4 - 4*b3 + 2*b2 + 2*b1 + 10) / 4 $$\nu^{3}$$ $$=$$ $$( -3\beta_{7} - 4\beta_{6} + 28\beta_{5} + 21\beta_{4} - 16\beta_{3} + 10\beta_{2} ) / 8$$ (-3*b7 - 4*b6 + 28*b5 + 21*b4 - 16*b3 + 10*b2) / 8 $$\nu^{4}$$ $$=$$ $$( 8\beta_{7} + 21\beta_{6} - 33\beta_{5} - 24\beta_{4} + 16\beta_{3} - 32\beta_{2} + 66\beta _1 - 146 ) / 4$$ (8*b7 + 21*b6 - 33*b5 - 24*b4 + 16*b3 - 32*b2 + 66*b1 - 146) / 4 $$\nu^{5}$$ $$=$$ $$( -25\beta_{7} + 6\beta_{6} + 112\beta_{5} + 175\beta_{4} - 376\beta_{3} + 502\beta_{2} - 522\beta _1 + 1500 ) / 8$$ (-25*b7 + 6*b6 + 112*b5 + 175*b4 - 376*b3 + 502*b2 - 522*b1 + 1500) / 8 $$\nu^{6}$$ $$=$$ $$( -16\beta_{7} + 24\beta_{6} + 93\beta_{5} + 148\beta_{4} + 804\beta_{3} - 1068\beta_{2} + 743\beta _1 - 2948 ) / 4$$ (-16*b7 + 24*b6 + 93*b5 + 148*b4 + 804*b3 - 1068*b2 + 743*b1 - 2948) / 4 $$\nu^{7}$$ $$=$$ $$( 693 \beta_{7} + 1306 \beta_{6} - 2864 \beta_{5} - 2751 \beta_{4} - 3876 \beta_{3} + 5654 \beta_{2} - 4048 \beta _1 + 18900 ) / 8$$ (693*b7 + 1306*b6 - 2864*b5 - 2751*b4 - 3876*b3 + 5654*b2 - 4048*b1 + 18900) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −0.639946 − 0.719687i −0.639946 + 0.719687i 1.52814 − 1.99551i 1.52814 + 1.99551i −3.26020 + 1.99551i −3.26020 − 1.99551i 2.37200 + 0.719687i 2.37200 − 0.719687i
0 −4.11439 0 1.10245 0 3.73205 5.92214i 0 7.92820 0
209.2 0 −4.11439 0 1.10245 0 3.73205 + 5.92214i 0 7.92820 0
209.3 0 −1.75265 0 6.54099 0 0.267949 6.99487i 0 −5.92820 0
209.4 0 −1.75265 0 6.54099 0 0.267949 + 6.99487i 0 −5.92820 0
209.5 0 1.75265 0 −6.54099 0 0.267949 6.99487i 0 −5.92820 0
209.6 0 1.75265 0 −6.54099 0 0.267949 + 6.99487i 0 −5.92820 0
209.7 0 4.11439 0 −1.10245 0 3.73205 5.92214i 0 7.92820 0
209.8 0 4.11439 0 −1.10245 0 3.73205 + 5.92214i 0 7.92820 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.h.d 8
3.b odd 2 1 2016.3.l.f 8
4.b odd 2 1 56.3.h.d 8
7.b odd 2 1 inner 224.3.h.d 8
8.b even 2 1 inner 224.3.h.d 8
8.d odd 2 1 56.3.h.d 8
12.b even 2 1 504.3.l.f 8
21.c even 2 1 2016.3.l.f 8
24.f even 2 1 504.3.l.f 8
24.h odd 2 1 2016.3.l.f 8
28.d even 2 1 56.3.h.d 8
28.f even 6 2 392.3.j.d 16
28.g odd 6 2 392.3.j.d 16
56.e even 2 1 56.3.h.d 8
56.h odd 2 1 inner 224.3.h.d 8
56.k odd 6 2 392.3.j.d 16
56.m even 6 2 392.3.j.d 16
84.h odd 2 1 504.3.l.f 8
168.e odd 2 1 504.3.l.f 8
168.i even 2 1 2016.3.l.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.d 8 4.b odd 2 1
56.3.h.d 8 8.d odd 2 1
56.3.h.d 8 28.d even 2 1
56.3.h.d 8 56.e even 2 1
224.3.h.d 8 1.a even 1 1 trivial
224.3.h.d 8 7.b odd 2 1 inner
224.3.h.d 8 8.b even 2 1 inner
224.3.h.d 8 56.h odd 2 1 inner
392.3.j.d 16 28.f even 6 2
392.3.j.d 16 28.g odd 6 2
392.3.j.d 16 56.k odd 6 2
392.3.j.d 16 56.m even 6 2
504.3.l.f 8 12.b even 2 1
504.3.l.f 8 24.f even 2 1
504.3.l.f 8 84.h odd 2 1
504.3.l.f 8 168.e odd 2 1
2016.3.l.f 8 3.b odd 2 1
2016.3.l.f 8 21.c even 2 1
2016.3.l.f 8 24.h odd 2 1
2016.3.l.f 8 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 20T_{3}^{2} + 52$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - 20 T^{2} + 52)^{2}$$
$5$ $$(T^{4} - 44 T^{2} + 52)^{2}$$
$7$ $$(T^{4} - 8 T^{3} + 102 T^{2} - 392 T + 2401)^{2}$$
$11$ $$(T^{4} + 120 T^{2} + 528)^{2}$$
$13$ $$(T^{4} - 332 T^{2} + 27508)^{2}$$
$17$ $$(T^{4} + 1008 T^{2} + 247104)^{2}$$
$19$ $$(T^{4} - 788 T^{2} + 114868)^{2}$$
$23$ $$(T^{2} - 8 T - 416)^{4}$$
$29$ $$(T^{4} + 1920 T^{2} + 33792)^{2}$$
$31$ $$(T^{4} + 3984 T^{2} + 3322176)^{2}$$
$37$ $$(T^{4} + 864 T^{2} + 76032)^{2}$$
$41$ $$(T^{4} + 1344 T^{2} + 439296)^{2}$$
$43$ $$(T^{4} + 6072 T^{2} + 7730448)^{2}$$
$47$ $$(T^{4} + 3984 T^{2} + 3322176)^{2}$$
$53$ $$(T^{4} + 3456 T^{2} + 2737152)^{2}$$
$59$ $$(T^{4} - 4724 T^{2} + 5029492)^{2}$$
$61$ $$(T^{4} - 44 T^{2} + 52)^{2}$$
$67$ $$(T^{4} + 6072 T^{2} + 7730448)^{2}$$
$71$ $$(T^{2} + 136 T + 2596)^{4}$$
$73$ $$(T^{4} + 11952 T^{2} + 29899584)^{2}$$
$79$ $$(T^{2} - 8 T - 956)^{4}$$
$83$ $$(T^{4} - 20948 T^{2} + 114868)^{2}$$
$89$ $$(T^{4} + 14256 T^{2} + 29899584)^{2}$$
$97$ $$(T^{4} + 24048 T^{2} + \cdots + 102163776)^{2}$$