# Properties

 Label 2025.2.b.h Level $2025$ Weight $2$ Character orbit 2025.b Analytic conductor $16.170$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 405) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + (2 \beta_1 - 2) q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{7} + (2 \beta_{3} - 2 \beta_{2}) q^{8}+O(q^{10})$$ q - b3 * q^2 + (2*b1 - 2) * q^4 + (-b3 - 2*b2) * q^7 + (2*b3 - 2*b2) * q^8 $$q - \beta_{3} q^{2} + (2 \beta_1 - 2) q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{7} + (2 \beta_{3} - 2 \beta_{2}) q^{8} + (\beta_1 + 4) q^{11} - 2 \beta_{3} q^{13} - 2 \beta_1 q^{14} + ( - 4 \beta_1 + 8) q^{16} + (\beta_{3} + \beta_{2}) q^{17} + ( - 2 \beta_1 - 1) q^{19} + ( - 3 \beta_{3} - \beta_{2}) q^{22} + ( - 2 \beta_{3} - \beta_{2}) q^{23} + (4 \beta_1 - 8) q^{26} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{28} + (3 \beta_1 - 2) q^{29} - 3 q^{31} - 8 \beta_{3} q^{32} + 2 q^{34} + ( - \beta_{3} - \beta_{2}) q^{37} + ( - \beta_{3} + 2 \beta_{2}) q^{38} + (3 \beta_1 + 2) q^{41} + ( - 3 \beta_{3} + \beta_{2}) q^{43} + (6 \beta_1 - 2) q^{44} + (2 \beta_1 - 6) q^{46} + (\beta_{3} + 4 \beta_{2}) q^{47} + ( - 6 \beta_1 - 5) q^{49} + (8 \beta_{3} - 4 \beta_{2}) q^{52} + (\beta_{3} + 3 \beta_{2}) q^{53} - 12 q^{56} + (5 \beta_{3} - 3 \beta_{2}) q^{58} + (\beta_1 - 10) q^{59} + 4 q^{61} + 3 \beta_{3} q^{62} + (8 \beta_1 - 16) q^{64} + (2 \beta_{3} + \beta_{2}) q^{67} + 2 \beta_{2} q^{68} + (\beta_1 + 2) q^{71} + (5 \beta_{3} + 2 \beta_{2}) q^{73} - 2 q^{74} + (2 \beta_1 - 10) q^{76} + ( - 7 \beta_{3} - 11 \beta_{2}) q^{77} + ( - 2 \beta_1 - 12) q^{79} + (\beta_{3} - 3 \beta_{2}) q^{82} + 3 \beta_{3} q^{83} + (8 \beta_1 - 14) q^{86} + (2 \beta_{3} - 8 \beta_{2}) q^{88} + 3 \beta_1 q^{89} - 4 \beta_1 q^{91} + (4 \beta_{3} - 4 \beta_{2}) q^{92} + (6 \beta_1 - 4) q^{94} + ( - 5 \beta_{3} - 3 \beta_{2}) q^{97} + ( - \beta_{3} + 6 \beta_{2}) q^{98}+O(q^{100})$$ q - b3 * q^2 + (2*b1 - 2) * q^4 + (-b3 - 2*b2) * q^7 + (2*b3 - 2*b2) * q^8 + (b1 + 4) * q^11 - 2*b3 * q^13 - 2*b1 * q^14 + (-4*b1 + 8) * q^16 + (b3 + b2) * q^17 + (-2*b1 - 1) * q^19 + (-3*b3 - b2) * q^22 + (-2*b3 - b2) * q^23 + (4*b1 - 8) * q^26 + (-4*b3 - 2*b2) * q^28 + (3*b1 - 2) * q^29 - 3 * q^31 - 8*b3 * q^32 + 2 * q^34 + (-b3 - b2) * q^37 + (-b3 + 2*b2) * q^38 + (3*b1 + 2) * q^41 + (-3*b3 + b2) * q^43 + (6*b1 - 2) * q^44 + (2*b1 - 6) * q^46 + (b3 + 4*b2) * q^47 + (-6*b1 - 5) * q^49 + (8*b3 - 4*b2) * q^52 + (b3 + 3*b2) * q^53 - 12 * q^56 + (5*b3 - 3*b2) * q^58 + (b1 - 10) * q^59 + 4 * q^61 + 3*b3 * q^62 + (8*b1 - 16) * q^64 + (2*b3 + b2) * q^67 + 2*b2 * q^68 + (b1 + 2) * q^71 + (5*b3 + 2*b2) * q^73 - 2 * q^74 + (2*b1 - 10) * q^76 + (-7*b3 - 11*b2) * q^77 + (-2*b1 - 12) * q^79 + (b3 - 3*b2) * q^82 + 3*b3 * q^83 + (8*b1 - 14) * q^86 + (2*b3 - 8*b2) * q^88 + 3*b1 * q^89 - 4*b1 * q^91 + (4*b3 - 4*b2) * q^92 + (6*b1 - 4) * q^94 + (-5*b3 - 3*b2) * q^97 + (-b3 + 6*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4}+O(q^{10})$$ 4 * q - 8 * q^4 $$4 q - 8 q^{4} + 16 q^{11} + 32 q^{16} - 4 q^{19} - 32 q^{26} - 8 q^{29} - 12 q^{31} + 8 q^{34} + 8 q^{41} - 8 q^{44} - 24 q^{46} - 20 q^{49} - 48 q^{56} - 40 q^{59} + 16 q^{61} - 64 q^{64} + 8 q^{71} - 8 q^{74} - 40 q^{76} - 48 q^{79} - 56 q^{86} - 16 q^{94}+O(q^{100})$$ 4 * q - 8 * q^4 + 16 * q^11 + 32 * q^16 - 4 * q^19 - 32 * q^26 - 8 * q^29 - 12 * q^31 + 8 * q^34 + 8 * q^41 - 8 * q^44 - 24 * q^46 - 20 * q^49 - 48 * q^56 - 40 * q^59 + 16 * q^61 - 64 * q^64 + 8 * q^71 - 8 * q^74 - 40 * q^76 - 48 * q^79 - 56 * q^86 - 16 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1$$ -v^3 + 2*v^2 - 1
 $$\zeta_{12}$$ $$=$$ $$( \beta_{2} + 2\beta_1 ) / 4$$ (b2 + 2*b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} + 2 ) / 4$$ (2*b3 + b2 + 2) / 4 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{2} ) / 2$$ (b2) / 2

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
2.73205i 0 −5.46410 0 0 1.26795i 9.46410i 0 0
649.2 0.732051i 0 1.46410 0 0 4.73205i 2.53590i 0 0
649.3 0.732051i 0 1.46410 0 0 4.73205i 2.53590i 0 0
649.4 2.73205i 0 −5.46410 0 0 1.26795i 9.46410i 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.h 4
3.b odd 2 1 2025.2.b.g 4
5.b even 2 1 inner 2025.2.b.h 4
5.c odd 4 1 405.2.a.h yes 2
5.c odd 4 1 2025.2.a.g 2
15.d odd 2 1 2025.2.b.g 4
15.e even 4 1 405.2.a.g 2
15.e even 4 1 2025.2.a.m 2
20.e even 4 1 6480.2.a.bi 2
45.k odd 12 2 405.2.e.i 4
45.l even 12 2 405.2.e.l 4
60.l odd 4 1 6480.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 15.e even 4 1
405.2.a.h yes 2 5.c odd 4 1
405.2.e.i 4 45.k odd 12 2
405.2.e.l 4 45.l even 12 2
2025.2.a.g 2 5.c odd 4 1
2025.2.a.m 2 15.e even 4 1
2025.2.b.g 4 3.b odd 2 1
2025.2.b.g 4 15.d odd 2 1
2025.2.b.h 4 1.a even 1 1 trivial
2025.2.b.h 4 5.b even 2 1 inner
6480.2.a.bi 2 20.e even 4 1
6480.2.a.br 2 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 $$S_{2}^{\mathrm{new}}(2025, [\chi])$$:

 $$T_{2}^{4} + 8T_{2}^{2} + 4$$ T2^4 + 8*T2^2 + 4 $$T_{11}^{2} - 8T_{11} + 13$$ T11^2 - 8*T11 + 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 8T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24T^{2} + 36$$
$11$ $$(T^{2} - 8 T + 13)^{2}$$
$13$ $$T^{4} + 32T^{2} + 64$$
$17$ $$T^{4} + 8T^{2} + 4$$
$19$ $$(T^{2} + 2 T - 11)^{2}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$(T^{2} + 4 T - 23)^{2}$$
$31$ $$(T + 3)^{4}$$
$37$ $$T^{4} + 8T^{2} + 4$$
$41$ $$(T^{2} - 4 T - 23)^{2}$$
$43$ $$T^{4} + 104T^{2} + 4$$
$47$ $$T^{4} + 104T^{2} + 2116$$
$53$ $$T^{4} + 56T^{2} + 484$$
$59$ $$(T^{2} + 20 T + 97)^{2}$$
$61$ $$(T - 4)^{4}$$
$67$ $$(T^{2} + 12)^{2}$$
$71$ $$(T^{2} - 4 T + 1)^{2}$$
$73$ $$T^{4} + 152T^{2} + 5476$$
$79$ $$(T^{2} + 24 T + 132)^{2}$$
$83$ $$T^{4} + 72T^{2} + 324$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$T^{4} + 152T^{2} + 5476$$