# Properties

 Label 225.4.f.d Level $225$ Weight $4$ Character orbit 225.f Analytic conductor $13.275$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: 16.0.11007531417600000000.1 Defining polynomial: $$x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1$$ x^16 - 7*x^12 + 48*x^8 - 7*x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{16}\cdot 3^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{13} q^{2} + (\beta_{15} + \beta_{9}) q^{4} + (2 \beta_{14} + 7 \beta_{3}) q^{7} + (2 \beta_{12} - 2 \beta_{10}) q^{8}+O(q^{10})$$ q + b13 * q^2 + (b15 + b9) * q^4 + (2*b14 + 7*b3) * q^7 + (2*b12 - 2*b10) * q^8 $$q + \beta_{13} q^{2} + (\beta_{15} + \beta_{9}) q^{4} + (2 \beta_{14} + 7 \beta_{3}) q^{7} + (2 \beta_{12} - 2 \beta_{10}) q^{8} + (7 \beta_{6} + \beta_{4}) q^{11} - 21 \beta_{2} q^{13} + (17 \beta_{7} - 4 \beta_{5}) q^{14} + (6 \beta_1 + 26) q^{16} + ( - 21 \beta_{13} - 7 \beta_{11}) q^{17} + (14 \beta_{15} - 35 \beta_{9}) q^{19} + (5 \beta_{14} + 63 \beta_{3}) q^{22} + ( - 31 \beta_{12} - 23 \beta_{10}) q^{23} - 21 \beta_{6} q^{26} + (9 \beta_{8} - 97 \beta_{2}) q^{28} + ( - 3 \beta_{7} - 17 \beta_{5}) q^{29} + (14 \beta_1 + 91) q^{31} + (40 \beta_{13} - 28 \beta_{11}) q^{32} + ( - 7 \beta_{15} - 189 \beta_{9}) q^{34} + ( - 12 \beta_{14} + 14 \beta_{3}) q^{37} + ( - 35 \beta_{12} - 28 \beta_{10}) q^{38} + (14 \beta_{6} + 14 \beta_{4}) q^{41} + ( - 30 \beta_{8} - 77 \beta_{2}) q^{43} + (32 \beta_{7} - 18 \beta_{5}) q^{44} + ( - 77 \beta_1 - 279) q^{46} + (77 \beta_{13} + 7 \beta_{11}) q^{47} + ( - 84 \beta_{15} - 344 \beta_{9}) q^{49} + ( - 21 \beta_{14} - 21 \beta_{3}) q^{52} + (4 \beta_{12} - 40 \beta_{10}) q^{53} + ( - 6 \beta_{6} - 14 \beta_{4}) q^{56} + (31 \beta_{8} + 27 \beta_{2}) q^{58} + ( - 91 \beta_{7} + 7 \beta_{5}) q^{59} + (28 \beta_1 + 413) q^{61} + (161 \beta_{13} - 28 \beta_{11}) q^{62} + (48 \beta_{15} + 152 \beta_{9}) q^{64} + (14 \beta_{14} - 99 \beta_{3}) q^{67} + (56 \beta_{12} + 70 \beta_{10}) q^{68} + ( - 94 \beta_{6} + 18 \beta_{4}) q^{71} + ( - 28 \beta_{8} + 154 \beta_{2}) q^{73} + ( - 46 \beta_{7} + 24 \beta_{5}) q^{74} + (21 \beta_1 - 595) q^{76} + ( - 297 \beta_{13} + 93 \beta_{11}) q^{77} + (28 \beta_{15} + 638 \beta_{9}) q^{79} + ( - 14 \beta_{14} + 126 \beta_{3}) q^{82} + ( - 231 \beta_{12} + 77 \beta_{10}) q^{83} + (73 \beta_{6} - 60 \beta_{4}) q^{86} + (28 \beta_{8} + 216 \beta_{2}) q^{88} + (154 \beta_{7} + 70 \beta_{5}) q^{89} + (126 \beta_1 + 441) q^{91} + ( - 416 \beta_{13} - 30 \beta_{11}) q^{92} + (63 \beta_{15} + 693 \beta_{9}) q^{94} + (56 \beta_{14} - 679 \beta_{3}) q^{97} + (764 \beta_{12} + 168 \beta_{10}) q^{98}+O(q^{100})$$ q + b13 * q^2 + (b15 + b9) * q^4 + (2*b14 + 7*b3) * q^7 + (2*b12 - 2*b10) * q^8 + (7*b6 + b4) * q^11 - 21*b2 * q^13 + (17*b7 - 4*b5) * q^14 + (6*b1 + 26) * q^16 + (-21*b13 - 7*b11) * q^17 + (14*b15 - 35*b9) * q^19 + (5*b14 + 63*b3) * q^22 + (-31*b12 - 23*b10) * q^23 - 21*b6 * q^26 + (9*b8 - 97*b2) * q^28 + (-3*b7 - 17*b5) * q^29 + (14*b1 + 91) * q^31 + (40*b13 - 28*b11) * q^32 + (-7*b15 - 189*b9) * q^34 + (-12*b14 + 14*b3) * q^37 + (-35*b12 - 28*b10) * q^38 + (14*b6 + 14*b4) * q^41 + (-30*b8 - 77*b2) * q^43 + (32*b7 - 18*b5) * q^44 + (-77*b1 - 279) * q^46 + (77*b13 + 7*b11) * q^47 + (-84*b15 - 344*b9) * q^49 + (-21*b14 - 21*b3) * q^52 + (4*b12 - 40*b10) * q^53 + (-6*b6 - 14*b4) * q^56 + (31*b8 + 27*b2) * q^58 + (-91*b7 + 7*b5) * q^59 + (28*b1 + 413) * q^61 + (161*b13 - 28*b11) * q^62 + (48*b15 + 152*b9) * q^64 + (14*b14 - 99*b3) * q^67 + (56*b12 + 70*b10) * q^68 + (-94*b6 + 18*b4) * q^71 + (-28*b8 + 154*b2) * q^73 + (-46*b7 + 24*b5) * q^74 + (21*b1 - 595) * q^76 + (-297*b13 + 93*b11) * q^77 + (28*b15 + 638*b9) * q^79 + (-14*b14 + 126*b3) * q^82 + (-231*b12 + 77*b10) * q^83 + (73*b6 - 60*b4) * q^86 + (28*b8 + 216*b2) * q^88 + (154*b7 + 70*b5) * q^89 + (126*b1 + 441) * q^91 + (-416*b13 - 30*b11) * q^92 + (63*b15 + 693*b9) * q^94 + (56*b14 - 679*b3) * q^97 + (764*b12 + 168*b10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q + 416 q^{16} + 1456 q^{31} - 4464 q^{46} + 6608 q^{61} - 9520 q^{76} + 7056 q^{91}+O(q^{100})$$ 16 * q + 416 * q^16 + 1456 * q^31 - 4464 * q^46 + 6608 * q^61 - 9520 * q^76 + 7056 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{12} + 161 ) / 24$$ (v^12 + 161) / 24 $$\beta_{2}$$ $$=$$ $$( 3\nu^{13} - 20\nu^{9} + 136\nu^{5} + 19\nu ) / 12$$ (3*v^13 - 20*v^9 + 136*v^5 + 19*v) / 12 $$\beta_{3}$$ $$=$$ $$( 15\nu^{15} - 104\nu^{11} + 712\nu^{7} - 53\nu^{3} ) / 12$$ (15*v^15 - 104*v^11 + 712*v^7 - 53*v^3) / 12 $$\beta_{4}$$ $$=$$ $$( 3\nu^{15} + 47\nu^{13} - 336\nu^{9} + 2256\nu^{5} + 843\nu^{3} - 329\nu ) / 48$$ (3*v^15 + 47*v^13 - 336*v^9 + 2256*v^5 + 843*v^3 - 329*v) / 48 $$\beta_{5}$$ $$=$$ $$( -3\nu^{15} + 47\nu^{13} - 336\nu^{9} + 2256\nu^{5} - 843\nu^{3} - 329\nu ) / 48$$ (-3*v^15 + 47*v^13 - 336*v^9 + 2256*v^5 - 843*v^3 - 329*v) / 48 $$\beta_{6}$$ $$=$$ $$( -3\nu^{15} - 55\nu^{13} + 384\nu^{9} - 2640\nu^{5} - 987\nu^{3} + 385\nu ) / 48$$ (-3*v^15 - 55*v^13 + 384*v^9 - 2640*v^5 - 987*v^3 + 385*v) / 48 $$\beta_{7}$$ $$=$$ $$( 3\nu^{15} - 55\nu^{13} + 384\nu^{9} - 2640\nu^{5} + 987\nu^{3} + 385\nu ) / 48$$ (3*v^15 - 55*v^13 + 384*v^9 - 2640*v^5 + 987*v^3 + 385*v) / 48 $$\beta_{8}$$ $$=$$ $$( -13\nu^{13} + 88\nu^{9} - 608\nu^{5} - 85\nu ) / 8$$ (-13*v^13 + 88*v^9 - 608*v^5 - 85*v) / 8 $$\beta_{9}$$ $$=$$ $$( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 18$$ (7*v^14 - 48*v^10 + 330*v^6 - v^2) / 18 $$\beta_{10}$$ $$=$$ $$( 25\nu^{14} + 19\nu^{12} - 176\nu^{10} - 128\nu^{8} + 1216\nu^{6} + 928\nu^{4} - 351\nu^{2} - 69 ) / 48$$ (25*v^14 + 19*v^12 - 176*v^10 - 128*v^8 + 1216*v^6 + 928*v^4 - 351*v^2 - 69) / 48 $$\beta_{11}$$ $$=$$ $$( -25\nu^{14} + 19\nu^{12} + 176\nu^{10} - 128\nu^{8} - 1216\nu^{6} + 928\nu^{4} + 351\nu^{2} - 69 ) / 48$$ (-25*v^14 + 19*v^12 + 176*v^10 - 128*v^8 - 1216*v^6 + 928*v^4 + 351*v^2 - 69) / 48 $$\beta_{12}$$ $$=$$ $$( 29\nu^{14} + 23\nu^{12} - 208\nu^{10} - 160\nu^{8} + 1424\nu^{6} + 1088\nu^{4} - 411\nu^{2} - 81 ) / 48$$ (29*v^14 + 23*v^12 - 208*v^10 - 160*v^8 + 1424*v^6 + 1088*v^4 - 411*v^2 - 81) / 48 $$\beta_{13}$$ $$=$$ $$( 29\nu^{14} - 23\nu^{12} - 208\nu^{10} + 160\nu^{8} + 1424\nu^{6} - 1088\nu^{4} - 411\nu^{2} + 81 ) / 48$$ (29*v^14 - 23*v^12 - 208*v^10 + 160*v^8 + 1424*v^6 - 1088*v^4 - 411*v^2 + 81) / 48 $$\beta_{14}$$ $$=$$ $$( 67\nu^{15} - 464\nu^{11} + 3184\nu^{7} - 237\nu^{3} ) / 8$$ (67*v^15 - 464*v^11 + 3184*v^7 - 237*v^3) / 8 $$\beta_{15}$$ $$=$$ $$( 21\nu^{14} - 144\nu^{10} + 984\nu^{6} - 3\nu^{2} ) / 8$$ (21*v^14 - 144*v^10 + 984*v^6 - 3*v^2) / 8
 $$\nu$$ $$=$$ $$( -3\beta_{8} + 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - 9\beta_{2} ) / 36$$ (-3*b8 + 2*b7 + 2*b6 + b5 + b4 - 9*b2) / 36 $$\nu^{2}$$ $$=$$ $$( -\beta_{15} + \beta_{13} + \beta_{12} + 2\beta_{11} - 2\beta_{10} + 9\beta_{9} ) / 12$$ (-b15 + b13 + b12 + 2*b11 - 2*b10 + 9*b9) / 12 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} ) / 6$$ (b7 - b6 + b5 - b4) / 6 $$\nu^{4}$$ $$=$$ $$( 4\beta_{13} - 4\beta_{12} + 5\beta_{11} + 5\beta_{10} - 3\beta _1 + 21 ) / 12$$ (4*b13 - 4*b12 + 5*b11 + 5*b10 - 3*b1 + 21) / 12 $$\nu^{5}$$ $$=$$ $$( -15\beta_{8} - 7\beta_{7} - 7\beta_{6} - 8\beta_{5} - 8\beta_{4} - 99\beta_{2} ) / 36$$ (-15*b8 - 7*b7 - 7*b6 - 8*b5 - 8*b4 - 99*b2) / 36 $$\nu^{6}$$ $$=$$ $$( -4\beta_{15} + 27\beta_{9} ) / 3$$ (-4*b15 + 27*b9) / 3 $$\nu^{7}$$ $$=$$ $$( 13\beta_{14} + 6\beta_{7} - 6\beta_{6} + 7\beta_{5} - 7\beta_{4} - 87\beta_{3} ) / 12$$ (13*b14 + 6*b7 - 6*b6 + 7*b5 - 7*b4 - 87*b3) / 12 $$\nu^{8}$$ $$=$$ $$( 29\beta_{13} - 29\beta_{12} + 34\beta_{11} + 34\beta_{10} + 21\beta _1 - 141 ) / 12$$ (29*b13 - 29*b12 + 34*b11 + 34*b10 + 21*b1 - 141) / 12 $$\nu^{9}$$ $$=$$ $$( -47\beta_{7} - 47\beta_{6} - 55\beta_{5} - 55\beta_{4} ) / 18$$ (-47*b7 - 47*b6 - 55*b5 - 55*b4) / 18 $$\nu^{10}$$ $$=$$ $$( -55\beta_{15} - 76\beta_{13} - 76\beta_{12} - 89\beta_{11} + 89\beta_{10} + 369\beta_{9} ) / 12$$ (-55*b15 - 76*b13 - 76*b12 - 89*b11 + 89*b10 + 369*b9) / 12 $$\nu^{11}$$ $$=$$ $$( 89\beta_{14} - 41\beta_{7} + 41\beta_{6} - 48\beta_{5} + 48\beta_{4} - 597\beta_{3} ) / 12$$ (89*b14 - 41*b7 + 41*b6 - 48*b5 + 48*b4 - 597*b3) / 12 $$\nu^{12}$$ $$=$$ $$24\beta _1 - 161$$ 24*b1 - 161 $$\nu^{13}$$ $$=$$ $$( 699\beta_{8} - 322\beta_{7} - 322\beta_{6} - 377\beta_{5} - 377\beta_{4} + 4689\beta_{2} ) / 36$$ (699*b8 - 322*b7 - 322*b6 - 377*b5 - 377*b4 + 4689*b2) / 36 $$\nu^{14}$$ $$=$$ $$( 377\beta_{15} - 521\beta_{13} - 521\beta_{12} - 610\beta_{11} + 610\beta_{10} - 2529\beta_{9} ) / 12$$ (377*b15 - 521*b13 - 521*b12 - 610*b11 + 610*b10 - 2529*b9) / 12 $$\nu^{15}$$ $$=$$ $$( -281\beta_{7} + 281\beta_{6} - 329\beta_{5} + 329\beta_{4} ) / 6$$ (-281*b7 + 281*b6 - 329*b5 + 329*b4) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.596975 − 0.159959i −0.596975 + 0.159959i 0.418778 − 1.56290i −0.418778 + 1.56290i 1.56290 − 0.418778i −1.56290 + 0.418778i 0.159959 − 0.596975i −0.159959 + 0.596975i 0.596975 + 0.159959i −0.596975 − 0.159959i 0.418778 + 1.56290i −0.418778 − 1.56290i 1.56290 + 0.418778i −1.56290 − 0.418778i 0.159959 + 0.596975i −0.159959 − 0.596975i
−2.80252 + 2.80252i 0 7.70820i 0 0 −25.0049 25.0049i −0.817763 0.817763i 0 0
107.2 −2.80252 + 2.80252i 0 7.70820i 0 0 25.0049 + 25.0049i −0.817763 0.817763i 0 0
107.3 −1.07047 + 1.07047i 0 5.70820i 0 0 −7.85846 7.85846i −14.6742 14.6742i 0 0
107.4 −1.07047 + 1.07047i 0 5.70820i 0 0 7.85846 + 7.85846i −14.6742 14.6742i 0 0
107.5 1.07047 1.07047i 0 5.70820i 0 0 −7.85846 7.85846i 14.6742 + 14.6742i 0 0
107.6 1.07047 1.07047i 0 5.70820i 0 0 7.85846 + 7.85846i 14.6742 + 14.6742i 0 0
107.7 2.80252 2.80252i 0 7.70820i 0 0 −25.0049 25.0049i 0.817763 + 0.817763i 0 0
107.8 2.80252 2.80252i 0 7.70820i 0 0 25.0049 + 25.0049i 0.817763 + 0.817763i 0 0
143.1 −2.80252 2.80252i 0 7.70820i 0 0 −25.0049 + 25.0049i −0.817763 + 0.817763i 0 0
143.2 −2.80252 2.80252i 0 7.70820i 0 0 25.0049 25.0049i −0.817763 + 0.817763i 0 0
143.3 −1.07047 1.07047i 0 5.70820i 0 0 −7.85846 + 7.85846i −14.6742 + 14.6742i 0 0
143.4 −1.07047 1.07047i 0 5.70820i 0 0 7.85846 7.85846i −14.6742 + 14.6742i 0 0
143.5 1.07047 + 1.07047i 0 5.70820i 0 0 −7.85846 + 7.85846i 14.6742 14.6742i 0 0
143.6 1.07047 + 1.07047i 0 5.70820i 0 0 7.85846 7.85846i 14.6742 14.6742i 0 0
143.7 2.80252 + 2.80252i 0 7.70820i 0 0 −25.0049 + 25.0049i 0.817763 0.817763i 0 0
143.8 2.80252 + 2.80252i 0 7.70820i 0 0 25.0049 25.0049i 0.817763 0.817763i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 143.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.f.d 16
3.b odd 2 1 inner 225.4.f.d 16
5.b even 2 1 inner 225.4.f.d 16
5.c odd 4 2 inner 225.4.f.d 16
15.d odd 2 1 inner 225.4.f.d 16
15.e even 4 2 inner 225.4.f.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.f.d 16 1.a even 1 1 trivial
225.4.f.d 16 3.b odd 2 1 inner
225.4.f.d 16 5.b even 2 1 inner
225.4.f.d 16 5.c odd 4 2 inner
225.4.f.d 16 15.d odd 2 1 inner
225.4.f.d 16 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{8} + 252T_{2}^{4} + 1296$$ T2^8 + 252*T2^4 + 1296 $$T_{7}^{8} + 1578978T_{7}^{4} + 23854493601$$ T7^8 + 1578978*T7^4 + 23854493601

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 252 T^{4} + 1296)^{2}$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$(T^{8} + 1578978 T^{4} + \cdots + 23854493601)^{2}$$
$11$ $$(T^{4} + 2916 T^{2} + 2022084)^{4}$$
$13$ $$(T^{4} + 1750329)^{4}$$
$17$ $$(T^{8} + 90066312 T^{4} + \cdots + 973651921932816)^{2}$$
$19$ $$(T^{4} + 20090 T^{2} + 57684025)^{4}$$
$23$ $$(T^{8} + 2229304392 T^{4} + \cdots + 98\!\cdots\!96)^{2}$$
$29$ $$(T^{4} - 78516 T^{2} + \cdots + 451902564)^{4}$$
$31$ $$(T^{2} - 182 T - 539)^{8}$$
$37$ $$(T^{8} + 893687328 T^{4} + \cdots + 12\!\cdots\!16)^{2}$$
$41$ $$(T^{4} + 63504 T^{2} + 12446784)^{4}$$
$43$ $$(T^{8} + 56090700738 T^{4} + \cdots + 11\!\cdots\!61)^{2}$$
$47$ $$(T^{8} + 7295371272 T^{4} + \cdots + 63\!\cdots\!76)^{2}$$
$53$ $$(T^{8} + 17103117312 T^{4} + \cdots + 34\!\cdots\!56)^{2}$$
$59$ $$(T^{4} - 460404 T^{2} + \cdots + 7624433124)^{4}$$
$61$ $$(T^{2} - 826 T + 135289)^{8}$$
$67$ $$(T^{8} + 12465376578 T^{4} + \cdots + 75017234240001)^{2}$$
$71$ $$(T^{4} + 564624 T^{2} + \cdots + 501401664)^{4}$$
$73$ $$(T^{8} + 122891938848 T^{4} + \cdots + 14\!\cdots\!96)^{2}$$
$79$ $$(T^{4} + 884648 T^{2} + \cdots + 138208471696)^{4}$$
$83$ $$(T^{8} + 1318660873992 T^{4} + \cdots + 20\!\cdots\!96)^{2}$$
$89$ $$(T^{4} - 2603664 T^{2} + \cdots + 914104263744)^{4}$$
$97$ $$(T^{8} + 11211233284818 T^{4} + \cdots + 84\!\cdots\!61)^{2}$$