Properties

 Label 225.4.b.h Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ Defining polynomial: $$x^{4} - 9x^{2} + 25$$ x^4 - 9*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{2} + (2 \beta_{2} - 12) q^{4} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 6 \beta_{3} + 42 \beta_1) q^{8}+O(q^{10})$$ q + (b3 - b1) * q^2 + (2*b2 - 12) * q^4 + (-4*b3 - 13*b1) * q^7 + (-6*b3 + 42*b1) * q^8 $$q + (\beta_{3} - \beta_1) q^{2} + (2 \beta_{2} - 12) q^{4} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 6 \beta_{3} + 42 \beta_1) q^{8} + ( - 4 \beta_{2} - 14) q^{11} + ( - 16 \beta_{3} + 9 \beta_1) q^{13} + (9 \beta_{2} + 63) q^{14} + ( - 32 \beta_{2} + 60) q^{16} + ( - 20 \beta_{3} - 34 \beta_1) q^{17} + (4 \beta_{2} - 3) q^{19} + ( - 10 \beta_{3} - 62 \beta_1) q^{22} + (12 \beta_{3} - 66 \beta_1) q^{23} + ( - 25 \beta_{2} + 313) q^{26} + (22 \beta_{3} + 4 \beta_1) q^{28} + ( - 28 \beta_{2} + 46) q^{29} + (28 \beta_{2} + 61) q^{31} + (44 \beta_{3} - 332 \beta_1) q^{32} + (14 \beta_{2} + 346) q^{34} + ( - 24 \beta_{3} + 142 \beta_1) q^{37} + ( - 7 \beta_{3} + 79 \beta_1) q^{38} + (52 \beta_{2} - 196) q^{41} + ( - 4 \beta_{3} + 345 \beta_1) q^{43} + (20 \beta_{2} + 16) q^{44} + (78 \beta_{2} - 294) q^{46} + ( - 32 \beta_{3} - 310 \beta_1) q^{47} + ( - 104 \beta_{2} - 130) q^{49} + (210 \beta_{3} - 716 \beta_1) q^{52} + ( - 28 \beta_{3} + 424 \beta_1) q^{53} + (90 \beta_{2} + 90) q^{56} + (74 \beta_{3} - 578 \beta_1) q^{58} + (64 \beta_{2} + 62) q^{59} + (56 \beta_{2} + 375) q^{61} + (33 \beta_{3} + 471 \beta_1) q^{62} + (120 \beta_{2} - 688) q^{64} + ( - 100 \beta_{3} + 179 \beta_1) q^{67} + (172 \beta_{3} - 352 \beta_1) q^{68} + ( - 20 \beta_{2} - 412) q^{71} + (8 \beta_{3} - 54 \beta_1) q^{73} + ( - 166 \beta_{2} + 598) q^{74} + ( - 54 \beta_{2} + 188) q^{76} + (108 \beta_{3} + 486 \beta_1) q^{77} + ( - 160 \beta_{2} + 440) q^{79} + ( - 248 \beta_{3} + 1184 \beta_1) q^{82} + ( - 192 \beta_{3} - 78 \beta_1) q^{83} + ( - 349 \beta_{2} + 421) q^{86} + ( - 84 \beta_{3} - 132 \beta_1) q^{88} + ( - 144 \beta_{2} - 432) q^{89} + ( - 172 \beta_{2} - 1099) q^{91} + ( - 276 \beta_{3} + 1248 \beta_1) q^{92} + (278 \beta_{2} + 298) q^{94} - 521 \beta_1 q^{97} + ( - 26 \beta_{3} - 1846 \beta_1) q^{98}+O(q^{100})$$ q + (b3 - b1) * q^2 + (2*b2 - 12) * q^4 + (-4*b3 - 13*b1) * q^7 + (-6*b3 + 42*b1) * q^8 + (-4*b2 - 14) * q^11 + (-16*b3 + 9*b1) * q^13 + (9*b2 + 63) * q^14 + (-32*b2 + 60) * q^16 + (-20*b3 - 34*b1) * q^17 + (4*b2 - 3) * q^19 + (-10*b3 - 62*b1) * q^22 + (12*b3 - 66*b1) * q^23 + (-25*b2 + 313) * q^26 + (22*b3 + 4*b1) * q^28 + (-28*b2 + 46) * q^29 + (28*b2 + 61) * q^31 + (44*b3 - 332*b1) * q^32 + (14*b2 + 346) * q^34 + (-24*b3 + 142*b1) * q^37 + (-7*b3 + 79*b1) * q^38 + (52*b2 - 196) * q^41 + (-4*b3 + 345*b1) * q^43 + (20*b2 + 16) * q^44 + (78*b2 - 294) * q^46 + (-32*b3 - 310*b1) * q^47 + (-104*b2 - 130) * q^49 + (210*b3 - 716*b1) * q^52 + (-28*b3 + 424*b1) * q^53 + (90*b2 + 90) * q^56 + (74*b3 - 578*b1) * q^58 + (64*b2 + 62) * q^59 + (56*b2 + 375) * q^61 + (33*b3 + 471*b1) * q^62 + (120*b2 - 688) * q^64 + (-100*b3 + 179*b1) * q^67 + (172*b3 - 352*b1) * q^68 + (-20*b2 - 412) * q^71 + (8*b3 - 54*b1) * q^73 + (-166*b2 + 598) * q^74 + (-54*b2 + 188) * q^76 + (108*b3 + 486*b1) * q^77 + (-160*b2 + 440) * q^79 + (-248*b3 + 1184*b1) * q^82 + (-192*b3 - 78*b1) * q^83 + (-349*b2 + 421) * q^86 + (-84*b3 - 132*b1) * q^88 + (-144*b2 - 432) * q^89 + (-172*b2 - 1099) * q^91 + (-276*b3 + 1248*b1) * q^92 + (278*b2 + 298) * q^94 - 521*b1 * q^97 + (-26*b3 - 1846*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 48 q^{4}+O(q^{10})$$ 4 * q - 48 * q^4 $$4 q - 48 q^{4} - 56 q^{11} + 252 q^{14} + 240 q^{16} - 12 q^{19} + 1252 q^{26} + 184 q^{29} + 244 q^{31} + 1384 q^{34} - 784 q^{41} + 64 q^{44} - 1176 q^{46} - 520 q^{49} + 360 q^{56} + 248 q^{59} + 1500 q^{61} - 2752 q^{64} - 1648 q^{71} + 2392 q^{74} + 752 q^{76} + 1760 q^{79} + 1684 q^{86} - 1728 q^{89} - 4396 q^{91} + 1192 q^{94}+O(q^{100})$$ 4 * q - 48 * q^4 - 56 * q^11 + 252 * q^14 + 240 * q^16 - 12 * q^19 + 1252 * q^26 + 184 * q^29 + 244 * q^31 + 1384 * q^34 - 784 * q^41 + 64 * q^44 - 1176 * q^46 - 520 * q^49 + 360 * q^56 + 248 * q^59 + 1500 * q^61 - 2752 * q^64 - 1648 * q^71 + 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 1684 * q^86 - 1728 * q^89 - 4396 * q^91 + 1192 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 4\nu ) / 5$$ (v^3 - 4*v) / 5 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 14\nu ) / 5$$ (-v^3 + 14*v) / 5 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 9$$ 2*v^2 - 9
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 9 ) / 2$$ (b3 + 9) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 7\beta_1$$ 2*b2 + 7*b1

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
 −2.17945 + 0.500000i 2.17945 − 0.500000i 2.17945 + 0.500000i −2.17945 − 0.500000i
5.35890i 0 −20.7178 0 0 4.43560i 68.1534i 0 0
199.2 3.35890i 0 −3.28220 0 0 30.4356i 15.8466i 0 0
199.3 3.35890i 0 −3.28220 0 0 30.4356i 15.8466i 0 0
199.4 5.35890i 0 −20.7178 0 0 4.43560i 68.1534i 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.h 4
3.b odd 2 1 75.4.b.c 4
5.b even 2 1 inner 225.4.b.h 4
5.c odd 4 1 225.4.a.j 2
5.c odd 4 1 225.4.a.n 2
12.b even 2 1 1200.4.f.v 4
15.d odd 2 1 75.4.b.c 4
15.e even 4 1 75.4.a.d 2
15.e even 4 1 75.4.a.e yes 2
60.h even 2 1 1200.4.f.v 4
60.l odd 4 1 1200.4.a.bl 2
60.l odd 4 1 1200.4.a.bu 2

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

Hecke kernels

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

 $$T_{2}^{4} + 40T_{2}^{2} + 324$$ T2^4 + 40*T2^2 + 324 $$T_{7}^{4} + 946T_{7}^{2} + 18225$$ T7^4 + 946*T7^2 + 18225 $$T_{11}^{2} + 28T_{11} - 108$$ T11^2 + 28*T11 - 108

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 40T^{2} + 324$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 946 T^{2} + 18225$$
$11$ $$(T^{2} + 28 T - 108)^{2}$$
$13$ $$T^{4} + 9890 T^{2} + \cdots + 22877089$$
$17$ $$T^{4} + 17512 T^{2} + \cdots + 41525136$$
$19$ $$(T^{2} + 6 T - 295)^{2}$$
$23$ $$T^{4} + 14184 T^{2} + \cdots + 2624400$$
$29$ $$(T^{2} - 92 T - 12780)^{2}$$
$31$ $$(T^{2} - 122 T - 11175)^{2}$$
$37$ $$T^{4} + 62216 T^{2} + \cdots + 85008400$$
$41$ $$(T^{2} + 392 T - 12960)^{2}$$
$43$ $$T^{4} + 238658 T^{2} + \cdots + 14094675841$$
$47$ $$T^{4} + 231112 T^{2} + \cdots + 5874302736$$
$53$ $$T^{4} + 389344 T^{2} + \cdots + 27185414400$$
$59$ $$(T^{2} - 124 T - 73980)^{2}$$
$61$ $$(T^{2} - 750 T + 81041)^{2}$$
$67$ $$T^{4} + 444082 T^{2} + \cdots + 24951045681$$
$71$ $$(T^{2} + 824 T + 162144)^{2}$$
$73$ $$T^{4} + 8264 T^{2} + \cdots + 2890000$$
$79$ $$(T^{2} - 880 T - 292800)^{2}$$
$83$ $$T^{4} + 1413000 T^{2} + \cdots + 482096926224$$
$89$ $$(T^{2} + 864 T - 207360)^{2}$$
$97$ $$(T^{2} + 271441)^{2}$$