Properties

 Label 2025.2.b.c Level $2025$ Weight $2$ Character orbit 2025.b Analytic conductor $16.170$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} - 3 i q^{7} + 3 i q^{8} +O(q^{10})$$ q + i * q^2 + q^4 - 3*i * q^7 + 3*i * q^8 $$q + i q^{2} + q^{4} - 3 i q^{7} + 3 i q^{8} - 2 q^{11} + 2 i q^{13} + 3 q^{14} - q^{16} + 4 i q^{17} + 8 q^{19} - 2 i q^{22} - 3 i q^{23} - 2 q^{26} - 3 i q^{28} + q^{29} + 5 i q^{32} - 4 q^{34} - 4 i q^{37} + 8 i q^{38} + 5 q^{41} + 8 i q^{43} - 2 q^{44} + 3 q^{46} + 7 i q^{47} - 2 q^{49} + 2 i q^{52} + 2 i q^{53} + 9 q^{56} + i q^{58} + 14 q^{59} + 7 q^{61} - 7 q^{64} - 3 i q^{67} + 4 i q^{68} + 2 q^{71} - 4 i q^{73} + 4 q^{74} + 8 q^{76} + 6 i q^{77} + 6 q^{79} + 5 i q^{82} - 9 i q^{83} - 8 q^{86} - 6 i q^{88} + 15 q^{89} + 6 q^{91} - 3 i q^{92} - 7 q^{94} + 2 i q^{97} - 2 i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 - 3*i * q^7 + 3*i * q^8 - 2 * q^11 + 2*i * q^13 + 3 * q^14 - q^16 + 4*i * q^17 + 8 * q^19 - 2*i * q^22 - 3*i * q^23 - 2 * q^26 - 3*i * q^28 + q^29 + 5*i * q^32 - 4 * q^34 - 4*i * q^37 + 8*i * q^38 + 5 * q^41 + 8*i * q^43 - 2 * q^44 + 3 * q^46 + 7*i * q^47 - 2 * q^49 + 2*i * q^52 + 2*i * q^53 + 9 * q^56 + i * q^58 + 14 * q^59 + 7 * q^61 - 7 * q^64 - 3*i * q^67 + 4*i * q^68 + 2 * q^71 - 4*i * q^73 + 4 * q^74 + 8 * q^76 + 6*i * q^77 + 6 * q^79 + 5*i * q^82 - 9*i * q^83 - 8 * q^86 - 6*i * q^88 + 15 * q^89 + 6 * q^91 - 3*i * q^92 - 7 * q^94 + 2*i * q^97 - 2*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} - 4 q^{11} + 6 q^{14} - 2 q^{16} + 16 q^{19} - 4 q^{26} + 2 q^{29} - 8 q^{34} + 10 q^{41} - 4 q^{44} + 6 q^{46} - 4 q^{49} + 18 q^{56} + 28 q^{59} + 14 q^{61} - 14 q^{64} + 4 q^{71} + 8 q^{74} + 16 q^{76} + 12 q^{79} - 16 q^{86} + 30 q^{89} + 12 q^{91} - 14 q^{94}+O(q^{100})$$ 2 * q + 2 * q^4 - 4 * q^11 + 6 * q^14 - 2 * q^16 + 16 * q^19 - 4 * q^26 + 2 * q^29 - 8 * q^34 + 10 * q^41 - 4 * q^44 + 6 * q^46 - 4 * q^49 + 18 * q^56 + 28 * q^59 + 14 * q^61 - 14 * q^64 + 4 * q^71 + 8 * q^74 + 16 * q^76 + 12 * q^79 - 16 * q^86 + 30 * q^89 + 12 * q^91 - 14 * q^94

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 0 0 3.00000i 3.00000i 0 0
649.2 1.00000i 0 1.00000 0 0 3.00000i 3.00000i 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 2025.2.b.c 2
3.b odd 2 1 2025.2.b.d 2
5.b even 2 1 inner 2025.2.b.c 2
5.c odd 4 1 405.2.a.e 1
5.c odd 4 1 2025.2.a.b 1
9.c even 3 2 225.2.k.a 4
9.d odd 6 2 675.2.k.a 4
15.d odd 2 1 2025.2.b.d 2
15.e even 4 1 405.2.a.b 1
15.e even 4 1 2025.2.a.e 1
20.e even 4 1 6480.2.a.k 1
45.h odd 6 2 675.2.k.a 4
45.j even 6 2 225.2.k.a 4
45.k odd 12 2 45.2.e.a 2
45.k odd 12 2 225.2.e.a 2
45.l even 12 2 135.2.e.a 2
45.l even 12 2 675.2.e.a 2
60.l odd 4 1 6480.2.a.x 1
180.v odd 12 2 2160.2.q.a 2
180.x even 12 2 720.2.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.a 2 45.k odd 12 2
135.2.e.a 2 45.l even 12 2
225.2.e.a 2 45.k odd 12 2
225.2.k.a 4 9.c even 3 2
225.2.k.a 4 45.j even 6 2
405.2.a.b 1 15.e even 4 1
405.2.a.e 1 5.c odd 4 1
675.2.e.a 2 45.l even 12 2
675.2.k.a 4 9.d odd 6 2
675.2.k.a 4 45.h odd 6 2
720.2.q.d 2 180.x even 12 2
2025.2.a.b 1 5.c odd 4 1
2025.2.a.e 1 15.e even 4 1
2025.2.b.c 2 1.a even 1 1 trivial
2025.2.b.c 2 5.b even 2 1 inner
2025.2.b.d 2 3.b odd 2 1
2025.2.b.d 2 15.d odd 2 1
2160.2.q.a 2 180.v odd 12 2
6480.2.a.k 1 20.e even 4 1
6480.2.a.x 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2025, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{11} + 2$$ T11 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T - 5)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 49$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 14)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$(T - 2)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 6)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 4$$