# Properties

 Label 225.12.b.a Level $225$ Weight $12$ Character orbit 225.b Analytic conductor $172.877$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$172.877215626$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 39 \beta q^{2} - 4036 q^{4} + 13880 \beta q^{7} - 77532 \beta q^{8} +O(q^{10})$$ q + 39*b * q^2 - 4036 * q^4 + 13880*b * q^7 - 77532*b * q^8 $$q + 39 \beta q^{2} - 4036 q^{4} + 13880 \beta q^{7} - 77532 \beta q^{8} - 637836 q^{11} + 383107 \beta q^{13} - 2165280 q^{14} + 3829264 q^{16} + 1542177 \beta q^{17} + 19511404 q^{19} - 24875604 \beta q^{22} - 7656180 \beta q^{23} - 59764692 q^{26} - 56019680 \beta q^{28} + 10751262 q^{29} - 50937400 q^{31} - 9444240 \beta q^{32} - 240579612 q^{34} - 332370415 \beta q^{37} + 760944756 \beta q^{38} - 898833450 q^{41} - 478973594 \beta q^{43} + 2574306096 q^{44} + 1194364080 q^{46} - 777870672 \beta q^{47} + 1206709143 q^{49} - 1546219852 \beta q^{52} - 1896208515 \beta q^{53} + 4304576640 q^{56} + 419299218 \beta q^{58} + 555306924 q^{59} + 4950420998 q^{61} - 1986558600 \beta q^{62} + 9315634112 q^{64} - 2646199642 \beta q^{67} - 6224226372 \beta q^{68} + 14831086248 q^{71} + 6985502605 \beta q^{73} + 51849784740 q^{74} - 78748026544 q^{76} - 8853163680 \beta q^{77} - 3720542360 q^{79} - 35054504550 \beta q^{82} - 4384227018 \beta q^{83} + 74719880664 q^{86} + 49452700752 \beta q^{88} - 25472769174 q^{89} - 21270100640 q^{91} + 30900342480 \beta q^{92} + 121347824832 q^{94} + 19546247423 \beta q^{97} + 47061656577 \beta q^{98} +O(q^{100})$$ q + 39*b * q^2 - 4036 * q^4 + 13880*b * q^7 - 77532*b * q^8 - 637836 * q^11 + 383107*b * q^13 - 2165280 * q^14 + 3829264 * q^16 + 1542177*b * q^17 + 19511404 * q^19 - 24875604*b * q^22 - 7656180*b * q^23 - 59764692 * q^26 - 56019680*b * q^28 + 10751262 * q^29 - 50937400 * q^31 - 9444240*b * q^32 - 240579612 * q^34 - 332370415*b * q^37 + 760944756*b * q^38 - 898833450 * q^41 - 478973594*b * q^43 + 2574306096 * q^44 + 1194364080 * q^46 - 777870672*b * q^47 + 1206709143 * q^49 - 1546219852*b * q^52 - 1896208515*b * q^53 + 4304576640 * q^56 + 419299218*b * q^58 + 555306924 * q^59 + 4950420998 * q^61 - 1986558600*b * q^62 + 9315634112 * q^64 - 2646199642*b * q^67 - 6224226372*b * q^68 + 14831086248 * q^71 + 6985502605*b * q^73 + 51849784740 * q^74 - 78748026544 * q^76 - 8853163680*b * q^77 - 3720542360 * q^79 - 35054504550*b * q^82 - 4384227018*b * q^83 + 74719880664 * q^86 + 49452700752*b * q^88 - 25472769174 * q^89 - 21270100640 * q^91 + 30900342480*b * q^92 + 121347824832 * q^94 + 19546247423*b * q^97 + 47061656577*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8072 q^{4}+O(q^{10})$$ 2 * q - 8072 * q^4 $$2 q - 8072 q^{4} - 1275672 q^{11} - 4330560 q^{14} + 7658528 q^{16} + 39022808 q^{19} - 119529384 q^{26} + 21502524 q^{29} - 101874800 q^{31} - 481159224 q^{34} - 1797666900 q^{41} + 5148612192 q^{44} + 2388728160 q^{46} + 2413418286 q^{49} + 8609153280 q^{56} + 1110613848 q^{59} + 9900841996 q^{61} + 18631268224 q^{64} + 29662172496 q^{71} + 103699569480 q^{74} - 157496053088 q^{76} - 7441084720 q^{79} + 149439761328 q^{86} - 50945538348 q^{89} - 42540201280 q^{91} + 242695649664 q^{94}+O(q^{100})$$ 2 * q - 8072 * q^4 - 1275672 * q^11 - 4330560 * q^14 + 7658528 * q^16 + 39022808 * q^19 - 119529384 * q^26 + 21502524 * q^29 - 101874800 * q^31 - 481159224 * q^34 - 1797666900 * q^41 + 5148612192 * q^44 + 2388728160 * q^46 + 2413418286 * q^49 + 8609153280 * q^56 + 1110613848 * q^59 + 9900841996 * q^61 + 18631268224 * q^64 + 29662172496 * q^71 + 103699569480 * q^74 - 157496053088 * q^76 - 7441084720 * q^79 + 149439761328 * q^86 - 50945538348 * q^89 - 42540201280 * q^91 + 242695649664 * q^94

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
78.0000i 0 −4036.00 0 0 27760.0i 155064.i 0 0
199.2 78.0000i 0 −4036.00 0 0 27760.0i 155064.i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.12.b.a 2
3.b odd 2 1 75.12.b.a 2
5.b even 2 1 inner 225.12.b.a 2
5.c odd 4 1 9.12.a.a 1
5.c odd 4 1 225.12.a.f 1
15.d odd 2 1 75.12.b.a 2
15.e even 4 1 3.12.a.a 1
15.e even 4 1 75.12.a.a 1
20.e even 4 1 144.12.a.l 1
45.k odd 12 2 81.12.c.e 2
45.l even 12 2 81.12.c.a 2
60.l odd 4 1 48.12.a.f 1
105.k odd 4 1 147.12.a.c 1
120.q odd 4 1 192.12.a.g 1
120.w even 4 1 192.12.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 15.e even 4 1
9.12.a.a 1 5.c odd 4 1
48.12.a.f 1 60.l odd 4 1
75.12.a.a 1 15.e even 4 1
75.12.b.a 2 3.b odd 2 1
75.12.b.a 2 15.d odd 2 1
81.12.c.a 2 45.l even 12 2
81.12.c.e 2 45.k odd 12 2
144.12.a.l 1 20.e even 4 1
147.12.a.c 1 105.k odd 4 1
192.12.a.g 1 120.q odd 4 1
192.12.a.q 1 120.w even 4 1
225.12.a.f 1 5.c odd 4 1
225.12.b.a 2 1.a even 1 1 trivial
225.12.b.a 2 5.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 $$S_{12}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 6084$$ T2^2 + 6084 $$T_{11} + 637836$$ T11 + 637836

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 6084$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 770617600$$
$11$ $$(T + 637836)^{2}$$
$13$ $$T^{2} + 587083893796$$
$17$ $$T^{2} + 9513239597316$$
$19$ $$(T - 19511404)^{2}$$
$23$ $$T^{2} + \cdots + 234468368769600$$
$29$ $$(T - 10751262)^{2}$$
$31$ $$(T + 50937400)^{2}$$
$37$ $$T^{2} + 44\!\cdots\!00$$
$41$ $$(T + 898833450)^{2}$$
$43$ $$T^{2} + 91\!\cdots\!44$$
$47$ $$T^{2} + 24\!\cdots\!36$$
$53$ $$T^{2} + 14\!\cdots\!00$$
$59$ $$(T - 555306924)^{2}$$
$61$ $$(T - 4950420998)^{2}$$
$67$ $$T^{2} + 28\!\cdots\!56$$
$71$ $$(T - 14831086248)^{2}$$
$73$ $$T^{2} + 19\!\cdots\!00$$
$79$ $$(T + 3720542360)^{2}$$
$83$ $$T^{2} + 76\!\cdots\!96$$
$89$ $$(T + 25472769174)^{2}$$
$97$ $$T^{2} + 15\!\cdots\!16$$