# Properties

 Label 225.4.b.d Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} - q^{4} -20 i q^{7} + 21 i q^{8} +O(q^{10})$$ $$q + 3 i q^{2} - q^{4} -20 i q^{7} + 21 i q^{8} + 24 q^{11} + 74 i q^{13} + 60 q^{14} -71 q^{16} + 54 i q^{17} + 124 q^{19} + 72 i q^{22} + 120 i q^{23} -222 q^{26} + 20 i q^{28} -78 q^{29} + 200 q^{31} -45 i q^{32} -162 q^{34} + 70 i q^{37} + 372 i q^{38} -330 q^{41} + 92 i q^{43} -24 q^{44} -360 q^{46} -24 i q^{47} -57 q^{49} -74 i q^{52} -450 i q^{53} + 420 q^{56} -234 i q^{58} + 24 q^{59} -322 q^{61} + 600 i q^{62} -433 q^{64} + 196 i q^{67} -54 i q^{68} + 288 q^{71} -430 i q^{73} -210 q^{74} -124 q^{76} -480 i q^{77} + 520 q^{79} -990 i q^{82} -156 i q^{83} -276 q^{86} + 504 i q^{88} + 1026 q^{89} + 1480 q^{91} -120 i q^{92} + 72 q^{94} + 286 i q^{97} -171 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + O(q^{10})$$ $$2 q - 2 q^{4} + 48 q^{11} + 120 q^{14} - 142 q^{16} + 248 q^{19} - 444 q^{26} - 156 q^{29} + 400 q^{31} - 324 q^{34} - 660 q^{41} - 48 q^{44} - 720 q^{46} - 114 q^{49} + 840 q^{56} + 48 q^{59} - 644 q^{61} - 866 q^{64} + 576 q^{71} - 420 q^{74} - 248 q^{76} + 1040 q^{79} - 552 q^{86} + 2052 q^{89} + 2960 q^{91} + 144 q^{94} + O(q^{100})$$

## 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
3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 0 0
199.2 3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 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 225.4.b.d 2
3.b odd 2 1 75.4.b.a 2
5.b even 2 1 inner 225.4.b.d 2
5.c odd 4 1 45.4.a.b 1
5.c odd 4 1 225.4.a.g 1
12.b even 2 1 1200.4.f.m 2
15.d odd 2 1 75.4.b.a 2
15.e even 4 1 15.4.a.b 1
15.e even 4 1 75.4.a.a 1
20.e even 4 1 720.4.a.r 1
35.f even 4 1 2205.4.a.c 1
45.k odd 12 2 405.4.e.k 2
45.l even 12 2 405.4.e.d 2
60.h even 2 1 1200.4.f.m 2
60.l odd 4 1 240.4.a.f 1
60.l odd 4 1 1200.4.a.o 1
105.k odd 4 1 735.4.a.i 1
120.q odd 4 1 960.4.a.l 1
120.w even 4 1 960.4.a.bi 1
165.l odd 4 1 1815.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 15.e even 4 1
45.4.a.b 1 5.c odd 4 1
75.4.a.a 1 15.e even 4 1
75.4.b.a 2 3.b odd 2 1
75.4.b.a 2 15.d odd 2 1
225.4.a.g 1 5.c odd 4 1
225.4.b.d 2 1.a even 1 1 trivial
225.4.b.d 2 5.b even 2 1 inner
240.4.a.f 1 60.l odd 4 1
405.4.e.d 2 45.l even 12 2
405.4.e.k 2 45.k odd 12 2
720.4.a.r 1 20.e even 4 1
735.4.a.i 1 105.k odd 4 1
960.4.a.l 1 120.q odd 4 1
960.4.a.bi 1 120.w even 4 1
1200.4.a.o 1 60.l odd 4 1
1200.4.f.m 2 12.b even 2 1
1200.4.f.m 2 60.h even 2 1
1815.4.a.a 1 165.l odd 4 1
2205.4.a.c 1 35.f even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 9$$ $$T_{7}^{2} + 400$$ $$T_{11} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$400 + T^{2}$$
$11$ $$( -24 + T )^{2}$$
$13$ $$5476 + T^{2}$$
$17$ $$2916 + T^{2}$$
$19$ $$( -124 + T )^{2}$$
$23$ $$14400 + T^{2}$$
$29$ $$( 78 + T )^{2}$$
$31$ $$( -200 + T )^{2}$$
$37$ $$4900 + T^{2}$$
$41$ $$( 330 + T )^{2}$$
$43$ $$8464 + T^{2}$$
$47$ $$576 + T^{2}$$
$53$ $$202500 + T^{2}$$
$59$ $$( -24 + T )^{2}$$
$61$ $$( 322 + T )^{2}$$
$67$ $$38416 + T^{2}$$
$71$ $$( -288 + T )^{2}$$
$73$ $$184900 + T^{2}$$
$79$ $$( -520 + T )^{2}$$
$83$ $$24336 + T^{2}$$
$89$ $$( -1026 + T )^{2}$$
$97$ $$81796 + T^{2}$$