# Properties

 Label 225.4.a.j Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2754297513$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta + 12) q^{4} + ( - 4 \beta - 13) q^{7} + (6 \beta - 42) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b + 12) * q^4 + (-4*b - 13) * q^7 + (6*b - 42) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta + 12) q^{4} + ( - 4 \beta - 13) q^{7} + (6 \beta - 42) q^{8} + ( - 4 \beta - 14) q^{11} + (16 \beta - 9) q^{13} + ( - 9 \beta - 63) q^{14} + ( - 32 \beta + 60) q^{16} + ( - 20 \beta - 34) q^{17} + ( - 4 \beta + 3) q^{19} + ( - 10 \beta - 62) q^{22} + ( - 12 \beta + 66) q^{23} + ( - 25 \beta + 313) q^{26} + ( - 22 \beta - 4) q^{28} + (28 \beta - 46) q^{29} + (28 \beta + 61) q^{31} + (44 \beta - 332) q^{32} + ( - 14 \beta - 346) q^{34} + ( - 24 \beta + 142) q^{37} + (7 \beta - 79) q^{38} + (52 \beta - 196) q^{41} + (4 \beta - 345) q^{43} + ( - 20 \beta - 16) q^{44} + (78 \beta - 294) q^{46} + ( - 32 \beta - 310) q^{47} + (104 \beta + 130) q^{49} + (210 \beta - 716) q^{52} + (28 \beta - 424) q^{53} + (90 \beta + 90) q^{56} + ( - 74 \beta + 578) q^{58} + ( - 64 \beta - 62) q^{59} + (56 \beta + 375) q^{61} + (33 \beta + 471) q^{62} + ( - 120 \beta + 688) q^{64} + ( - 100 \beta + 179) q^{67} + ( - 172 \beta + 352) q^{68} + ( - 20 \beta - 412) q^{71} + ( - 8 \beta + 54) q^{73} + (166 \beta - 598) q^{74} + ( - 54 \beta + 188) q^{76} + (108 \beta + 486) q^{77} + (160 \beta - 440) q^{79} + ( - 248 \beta + 1184) q^{82} + (192 \beta + 78) q^{83} + ( - 349 \beta + 421) q^{86} + (84 \beta + 132) q^{88} + (144 \beta + 432) q^{89} + ( - 172 \beta - 1099) q^{91} + ( - 276 \beta + 1248) q^{92} + ( - 278 \beta - 298) q^{94} - 521 q^{97} + (26 \beta + 1846) q^{98}+O(q^{100})$$ q + (b - 1) * q^2 + (-2*b + 12) * q^4 + (-4*b - 13) * q^7 + (6*b - 42) * q^8 + (-4*b - 14) * q^11 + (16*b - 9) * q^13 + (-9*b - 63) * q^14 + (-32*b + 60) * q^16 + (-20*b - 34) * q^17 + (-4*b + 3) * q^19 + (-10*b - 62) * q^22 + (-12*b + 66) * q^23 + (-25*b + 313) * q^26 + (-22*b - 4) * q^28 + (28*b - 46) * q^29 + (28*b + 61) * q^31 + (44*b - 332) * q^32 + (-14*b - 346) * q^34 + (-24*b + 142) * q^37 + (7*b - 79) * q^38 + (52*b - 196) * q^41 + (4*b - 345) * q^43 + (-20*b - 16) * q^44 + (78*b - 294) * q^46 + (-32*b - 310) * q^47 + (104*b + 130) * q^49 + (210*b - 716) * q^52 + (28*b - 424) * q^53 + (90*b + 90) * q^56 + (-74*b + 578) * q^58 + (-64*b - 62) * q^59 + (56*b + 375) * q^61 + (33*b + 471) * q^62 + (-120*b + 688) * q^64 + (-100*b + 179) * q^67 + (-172*b + 352) * q^68 + (-20*b - 412) * q^71 + (-8*b + 54) * q^73 + (166*b - 598) * q^74 + (-54*b + 188) * q^76 + (108*b + 486) * q^77 + (160*b - 440) * q^79 + (-248*b + 1184) * q^82 + (192*b + 78) * q^83 + (-349*b + 421) * q^86 + (84*b + 132) * q^88 + (144*b + 432) * q^89 + (-172*b - 1099) * q^91 + (-276*b + 1248) * q^92 + (-278*b - 298) * q^94 - 521 * q^97 + (26*b + 1846) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8 $$2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8} - 28 q^{11} - 18 q^{13} - 126 q^{14} + 120 q^{16} - 68 q^{17} + 6 q^{19} - 124 q^{22} + 132 q^{23} + 626 q^{26} - 8 q^{28} - 92 q^{29} + 122 q^{31} - 664 q^{32} - 692 q^{34} + 284 q^{37} - 158 q^{38} - 392 q^{41} - 690 q^{43} - 32 q^{44} - 588 q^{46} - 620 q^{47} + 260 q^{49} - 1432 q^{52} - 848 q^{53} + 180 q^{56} + 1156 q^{58} - 124 q^{59} + 750 q^{61} + 942 q^{62} + 1376 q^{64} + 358 q^{67} + 704 q^{68} - 824 q^{71} + 108 q^{73} - 1196 q^{74} + 376 q^{76} + 972 q^{77} - 880 q^{79} + 2368 q^{82} + 156 q^{83} + 842 q^{86} + 264 q^{88} + 864 q^{89} - 2198 q^{91} + 2496 q^{92} - 596 q^{94} - 1042 q^{97} + 3692 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8 - 28 * q^11 - 18 * q^13 - 126 * q^14 + 120 * q^16 - 68 * q^17 + 6 * q^19 - 124 * q^22 + 132 * q^23 + 626 * q^26 - 8 * q^28 - 92 * q^29 + 122 * q^31 - 664 * q^32 - 692 * q^34 + 284 * q^37 - 158 * q^38 - 392 * q^41 - 690 * q^43 - 32 * q^44 - 588 * q^46 - 620 * q^47 + 260 * q^49 - 1432 * q^52 - 848 * q^53 + 180 * q^56 + 1156 * q^58 - 124 * q^59 + 750 * q^61 + 942 * q^62 + 1376 * q^64 + 358 * q^67 + 704 * q^68 - 824 * q^71 + 108 * q^73 - 1196 * q^74 + 376 * q^76 + 972 * q^77 - 880 * q^79 + 2368 * q^82 + 156 * q^83 + 842 * q^86 + 264 * q^88 + 864 * q^89 - 2198 * q^91 + 2496 * q^92 - 596 * q^94 - 1042 * q^97 + 3692 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.35890 4.35890
−5.35890 0 20.7178 0 0 4.43560 −68.1534 0 0
1.2 3.35890 0 3.28220 0 0 −30.4356 −15.8466 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.a.j 2
3.b odd 2 1 75.4.a.e yes 2
5.b even 2 1 225.4.a.n 2
5.c odd 4 2 225.4.b.h 4
12.b even 2 1 1200.4.a.bu 2
15.d odd 2 1 75.4.a.d 2
15.e even 4 2 75.4.b.c 4
60.h even 2 1 1200.4.a.bl 2
60.l odd 4 2 1200.4.f.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 15.d odd 2 1
75.4.a.e yes 2 3.b odd 2 1
75.4.b.c 4 15.e even 4 2
225.4.a.j 2 1.a even 1 1 trivial
225.4.a.n 2 5.b even 2 1
225.4.b.h 4 5.c odd 4 2
1200.4.a.bl 2 60.h even 2 1
1200.4.a.bu 2 12.b even 2 1
1200.4.f.v 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} + 2T_{2} - 18$$ T2^2 + 2*T2 - 18 $$T_{7}^{2} + 26T_{7} - 135$$ T7^2 + 26*T7 - 135

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 18$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 26T - 135$$
$11$ $$T^{2} + 28T - 108$$
$13$ $$T^{2} + 18T - 4783$$
$17$ $$T^{2} + 68T - 6444$$
$19$ $$T^{2} - 6T - 295$$
$23$ $$T^{2} - 132T + 1620$$
$29$ $$T^{2} + 92T - 12780$$
$31$ $$T^{2} - 122T - 11175$$
$37$ $$T^{2} - 284T + 9220$$
$41$ $$T^{2} + 392T - 12960$$
$43$ $$T^{2} + 690T + 118721$$
$47$ $$T^{2} + 620T + 76644$$
$53$ $$T^{2} + 848T + 164880$$
$59$ $$T^{2} + 124T - 73980$$
$61$ $$T^{2} - 750T + 81041$$
$67$ $$T^{2} - 358T - 157959$$
$71$ $$T^{2} + 824T + 162144$$
$73$ $$T^{2} - 108T + 1700$$
$79$ $$T^{2} + 880T - 292800$$
$83$ $$T^{2} - 156T - 694332$$
$89$ $$T^{2} - 864T - 207360$$
$97$ $$(T + 521)^{2}$$