# Properties

 Label 1224.4.c.e Level $1224$ Weight $4$ Character orbit 1224.c Analytic conductor $72.218$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1224.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$72.2183378470$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152$$ x^8 + 95*x^6 + 756*x^4 + 1780*x^2 + 1152 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: no (minimal twist has level 136) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7}+O(q^{10})$$ q + b4 * q^5 + (-b4 - b2 - b1) * q^7 $$q + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2}) q^{11} + (\beta_{7} + 5) q^{13} + ( - \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{17} + (\beta_{7} - \beta_{5} - 2 \beta_{3} + 4) q^{19} + ( - \beta_{6} + 5 \beta_1) q^{23} + ( - \beta_{7} + 3 \beta_{5} - 63) q^{25} + (2 \beta_{6} - 5 \beta_{4} - 18 \beta_{2}) q^{29} + (3 \beta_{6} - 4 \beta_{4} + 14 \beta_{2} + 5 \beta_1) q^{31} + ( - 2 \beta_{7} - 8 \beta_{5} + 130) q^{35} + (\beta_{4} - 28 \beta_{2} + 4 \beta_1) q^{37} + (6 \beta_{6} + 16 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{41} + ( - \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 2) q^{43} + (3 \beta_{7} - \beta_{5} - 2 \beta_{3} - 42) q^{47} + (\beta_{7} + 10 \beta_{5} + 8 \beta_{3} - 132) q^{49} + (3 \beta_{7} + 7 \beta_{5} - 6 \beta_{3} - 60) q^{53} + (7 \beta_{7} - 5 \beta_{5} + 6 \beta_{3} + 174) q^{55} + ( - \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 10) q^{59} + ( - 6 \beta_{6} + 5 \beta_{4} - 74 \beta_{2} - 16 \beta_1) q^{61} + (10 \beta_{6} + 16 \beta_{4} - 78 \beta_{2} - 20 \beta_1) q^{65} + ( - 9 \beta_{7} + \beta_{5} - 6 \beta_{3} - 76) q^{67} + (6 \beta_{6} - 27 \beta_{4} + 31 \beta_{2} - 29 \beta_1) q^{71} + (6 \beta_{6} - 26 \beta_{4} - 6 \beta_{2} + 20 \beta_1) q^{73} + ( - 3 \beta_{7} - 2 \beta_{5} - 8 \beta_{3} - 211) q^{77} + ( - 11 \beta_{6} - 16 \beta_{4} - 64 \beta_{2} - 5 \beta_1) q^{79} + ( - 9 \beta_{7} + 15 \beta_{5} - 2 \beta_{3} - 298) q^{83} + (7 \beta_{7} - 2 \beta_{6} + 11 \beta_{5} - 19 \beta_{4} + 94 \beta_{2} + \cdots - 268) q^{85}+ \cdots + (16 \beta_{6} - 22 \beta_{4} + 140 \beta_{2} - 24 \beta_1) q^{97}+O(q^{100})$$ q + b4 * q^5 + (-b4 - b2 - b1) * q^7 + (-b6 - b4 + 2*b2) * q^11 + (b7 + 5) * q^13 + (-b6 + 2*b4 - b3 + b2 + 2*b1 - 4) * q^17 + (b7 - b5 - 2*b3 + 4) * q^19 + (-b6 + 5*b1) * q^23 + (-b7 + 3*b5 - 63) * q^25 + (2*b6 - 5*b4 - 18*b2) * q^29 + (3*b6 - 4*b4 + 14*b2 + 5*b1) * q^31 + (-2*b7 - 8*b5 + 130) * q^35 + (b4 - 28*b2 + 4*b1) * q^37 + (6*b6 + 16*b4 - 2*b2 - 4*b1) * q^41 + (-b7 + 3*b5 - 2*b3 + 2) * q^43 + (3*b7 - b5 - 2*b3 - 42) * q^47 + (b7 + 10*b5 + 8*b3 - 132) * q^49 + (3*b7 + 7*b5 - 6*b3 - 60) * q^53 + (7*b7 - 5*b5 + 6*b3 + 174) * q^55 + (-b7 + 3*b5 - 2*b3 + 10) * q^59 + (-6*b6 + 5*b4 - 74*b2 - 16*b1) * q^61 + (10*b6 + 16*b4 - 78*b2 - 20*b1) * q^65 + (-9*b7 + b5 - 6*b3 - 76) * q^67 + (6*b6 - 27*b4 + 31*b2 - 29*b1) * q^71 + (6*b6 - 26*b4 - 6*b2 + 20*b1) * q^73 + (-3*b7 - 2*b5 - 8*b3 - 211) * q^77 + (-11*b6 - 16*b4 - 64*b2 - 5*b1) * q^79 + (-9*b7 + 15*b5 - 2*b3 - 298) * q^83 + (7*b7 - 2*b6 + 11*b5 - 19*b4 + 94*b2 - 16*b1 - 268) * q^85 + (-b7 - 22*b5 - 8*b3 - 23) * q^89 + (b6 + 21*b4 + 139*b2 - 20*b1) * q^91 + (8*b6 + 12*b4 + 120*b2 - 32*b1) * q^95 + (16*b6 - 22*b4 + 140*b2 - 24*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100})$$ 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu$$ 4*v $$\beta_{2}$$ $$=$$ $$( -3\nu^{7} - 284\nu^{5} - 2159\nu^{3} - 3402\nu ) / 172$$ (-3*v^7 - 284*v^5 - 2159*v^3 - 3402*v) / 172 $$\beta_{3}$$ $$=$$ $$( 3\nu^{6} + 327\nu^{4} + 6158\nu^{2} + 17420 ) / 172$$ (3*v^6 + 327*v^4 + 6158*v^2 + 17420) / 172 $$\beta_{4}$$ $$=$$ $$( 59\nu^{7} + 5485\nu^{5} + 33588\nu^{3} + 47900\nu ) / 1032$$ (59*v^7 + 5485*v^5 + 33588*v^3 + 47900*v) / 1032 $$\beta_{5}$$ $$=$$ $$( -41\nu^{6} - 3781\nu^{4} - 20634\nu^{2} - 22844 ) / 172$$ (-41*v^6 - 3781*v^4 - 20634*v^2 - 22844) / 172 $$\beta_{6}$$ $$=$$ $$( -109\nu^{7} - 9989\nu^{5} - 48702\nu^{3} - 15160\nu ) / 1032$$ (-109*v^7 - 9989*v^5 - 48702*v^3 - 15160*v) / 1032 $$\beta_{7}$$ $$=$$ $$( 45\nu^{6} + 4217\nu^{4} + 28386\nu^{2} + 35292 ) / 172$$ (45*v^6 + 4217*v^4 + 28386*v^2 + 35292) / 172
 $$\nu$$ $$=$$ $$( \beta_1 ) / 4$$ (b1) / 4 $$\nu^{2}$$ $$=$$ $$( -3\beta_{7} - 3\beta_{5} + 4\beta_{3} - 188 ) / 8$$ (-3*b7 - 3*b5 + 4*b3 - 188) / 8 $$\nu^{3}$$ $$=$$ $$( 7\beta_{6} + 23\beta_{4} + 33\beta_{2} - 78\beta_1 ) / 4$$ (7*b6 + 23*b4 + 33*b2 - 78*b1) / 4 $$\nu^{4}$$ $$=$$ $$( 277\beta_{7} + 279\beta_{5} - 342\beta_{3} + 14856 ) / 8$$ (277*b7 + 279*b5 - 342*b3 + 14856) / 8 $$\nu^{5}$$ $$=$$ $$( -619\beta_{6} - 2075\beta_{4} - 3053\beta_{2} + 6708\beta_1 ) / 4$$ (-619*b6 - 2075*b4 - 3053*b2 + 6708*b1) / 4 $$\nu^{6}$$ $$=$$ $$( -24035\beta_{7} - 24253\beta_{5} + 29526\beta_{3} - 1279856 ) / 8$$ (-24035*b7 - 24253*b5 + 29526*b3 - 1279856) / 8 $$\nu^{7}$$ $$=$$ $$( 53561\beta_{6} + 179881\beta_{4} + 265039\beta_{2} - 580024\beta_1 ) / 4$$ (53561*b6 + 179881*b4 + 265039*b2 - 580024*b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$$.

 $$n$$ $$137$$ $$613$$ $$649$$ $$919$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.03229i − 2.20783i 9.30031i 1.60125i − 1.60125i − 9.30031i 2.20783i 1.03229i
0 0 0 18.2701i 0 13.8757i 0 0 0
577.2 0 0 0 16.4090i 0 34.5010i 0 0 0
577.3 0 0 0 11.5318i 0 23.0485i 0 0 0
577.4 0 0 0 4.89575i 0 4.46235i 0 0 0
577.5 0 0 0 4.89575i 0 4.46235i 0 0 0
577.6 0 0 0 11.5318i 0 23.0485i 0 0 0
577.7 0 0 0 16.4090i 0 34.5010i 0 0 0
577.8 0 0 0 18.2701i 0 13.8757i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.4.c.e 8
3.b odd 2 1 136.4.b.b 8
12.b even 2 1 272.4.b.f 8
17.b even 2 1 inner 1224.4.c.e 8
51.c odd 2 1 136.4.b.b 8
51.f odd 4 2 2312.4.a.k 8
204.h even 2 1 272.4.b.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.b 8 3.b odd 2 1
136.4.b.b 8 51.c odd 2 1
272.4.b.f 8 12.b even 2 1
272.4.b.f 8 204.h even 2 1
1224.4.c.e 8 1.a even 1 1 trivial
1224.4.c.e 8 17.b even 2 1 inner
2312.4.a.k 8 51.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1224, [\chi])$$:

 $$T_{5}^{8} + 760T_{5}^{6} + 187712T_{5}^{4} + 16028160T_{5}^{2} + 286466048$$ T5^8 + 760*T5^6 + 187712*T5^4 + 16028160*T5^2 + 286466048 $$T_{47}^{4} + 156T_{47}^{3} - 57344T_{47}^{2} - 3407872T_{47} + 134217728$$ T47^4 + 156*T47^3 - 57344*T47^2 - 3407872*T47 + 134217728

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 760 T^{6} + \cdots + 286466048$$
$7$ $$T^{8} + 1934 T^{6} + \cdots + 2424307712$$
$11$ $$T^{8} + 7406 T^{6} + \cdots + 1063158272$$
$13$ $$(T^{4} - 22 T^{3} - 5836 T^{2} + \cdots + 8525216)^{2}$$
$17$ $$T^{8} + \cdots + 582622237229761$$
$19$ $$(T^{4} - 24 T^{3} - 21056 T^{2} + \cdots + 44946176)^{2}$$
$23$ $$T^{8} + 36734 T^{6} + \cdots + 2935871578112$$
$29$ $$T^{8} + 106232 T^{6} + \cdots + 14\!\cdots\!52$$
$31$ $$T^{8} + 155918 T^{6} + \cdots + 32\!\cdots\!52$$
$37$ $$T^{8} + 166072 T^{6} + \cdots + 54\!\cdots\!12$$
$41$ $$T^{8} + 419392 T^{6} + \cdots + 55\!\cdots\!32$$
$43$ $$(T^{4} - 4 T^{3} - 60592 T^{2} + \cdots + 112195072)^{2}$$
$47$ $$(T^{4} + 156 T^{3} - 57344 T^{2} + \cdots + 134217728)^{2}$$
$53$ $$(T^{4} + 236 T^{3} + \cdots - 14761769616)^{2}$$
$59$ $$(T^{4} - 36 T^{3} - 60112 T^{2} + \cdots + 70465536)^{2}$$
$61$ $$T^{8} + 1597240 T^{6} + \cdots + 18\!\cdots\!28$$
$67$ $$(T^{4} + 312 T^{3} + \cdots + 30967766784)^{2}$$
$71$ $$T^{8} + 1607086 T^{6} + \cdots + 14\!\cdots\!28$$
$73$ $$T^{8} + 1779648 T^{6} + \cdots + 21\!\cdots\!68$$
$79$ $$T^{8} + 1578638 T^{6} + \cdots + 17\!\cdots\!72$$
$83$ $$(T^{4} + 1236 T^{3} + \cdots - 54225864704)^{2}$$
$89$ $$(T^{4} + 34 T^{3} - 1202908 T^{2} + \cdots + 21605388512)^{2}$$
$97$ $$T^{8} + 5185024 T^{6} + \cdots + 13\!\cdots\!92$$