Properties

Label 2025.4.a.n
Level $2025$
Weight $4$
Character orbit 2025.a
Self dual yes
Analytic conductor $119.479$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(119.478867762\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 3 \beta ) q^{4} + ( -2 - 3 \beta ) q^{7} + ( 17 - \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 3 \beta ) q^{4} + ( -2 - 3 \beta ) q^{7} + ( 17 - \beta ) q^{8} + ( -37 + 8 \beta ) q^{11} + ( -2 + 15 \beta ) q^{13} + ( -26 - 8 \beta ) q^{14} + ( 1 - 9 \beta ) q^{16} + ( 45 + 9 \beta ) q^{17} + ( -25 - 27 \beta ) q^{19} + ( 27 - 21 \beta ) q^{22} + ( 26 - 19 \beta ) q^{23} + ( 118 + 28 \beta ) q^{26} + ( -74 - 18 \beta ) q^{28} + ( 26 - \beta ) q^{29} + ( 20 + 3 \beta ) q^{31} + ( -207 - 9 \beta ) q^{32} + ( 117 + 63 \beta ) q^{34} + ( 52 - 54 \beta ) q^{37} + ( -241 - 79 \beta ) q^{38} + ( -17 - 98 \beta ) q^{41} + ( -47 + 6 \beta ) q^{43} + ( 155 - 79 \beta ) q^{44} + ( -126 - 12 \beta ) q^{46} + ( 154 + 91 \beta ) q^{47} + ( -267 + 21 \beta ) q^{49} + ( 358 + 54 \beta ) q^{52} + ( 108 - 162 \beta ) q^{53} + ( -10 - 46 \beta ) q^{56} + ( 18 + 24 \beta ) q^{58} + ( -467 + 136 \beta ) q^{59} + ( 272 - 105 \beta ) q^{61} + ( 44 + 26 \beta ) q^{62} + ( -287 - 153 \beta ) q^{64} + ( -461 - 66 \beta ) q^{67} + ( 261 + 171 \beta ) q^{68} + ( -612 - 144 \beta ) q^{71} + ( 349 - 243 \beta ) q^{73} + ( -380 - 56 \beta ) q^{74} + ( -673 - 183 \beta ) q^{76} + ( -118 + 71 \beta ) q^{77} + ( -556 + 309 \beta ) q^{79} + ( -801 - 213 \beta ) q^{82} + ( 460 - 107 \beta ) q^{83} + ( 1 - 35 \beta ) q^{86} + ( -693 + 165 \beta ) q^{88} + ( 234 - 72 \beta ) q^{89} + ( -356 - 69 \beta ) q^{91} + ( -430 + 2 \beta ) q^{92} + ( 882 + 336 \beta ) q^{94} + ( -317 - 102 \beta ) q^{97} + ( -99 - 225 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 5q^{4} - 7q^{7} + 33q^{8} + O(q^{10}) \) \( 2q + 3q^{2} + 5q^{4} - 7q^{7} + 33q^{8} - 66q^{11} + 11q^{13} - 60q^{14} - 7q^{16} + 99q^{17} - 77q^{19} + 33q^{22} + 33q^{23} + 264q^{26} - 166q^{28} + 51q^{29} + 43q^{31} - 423q^{32} + 297q^{34} + 50q^{37} - 561q^{38} - 132q^{41} - 88q^{43} + 231q^{44} - 264q^{46} + 399q^{47} - 513q^{49} + 770q^{52} + 54q^{53} - 66q^{56} + 60q^{58} - 798q^{59} + 439q^{61} + 114q^{62} - 727q^{64} - 988q^{67} + 693q^{68} - 1368q^{71} + 455q^{73} - 816q^{74} - 1529q^{76} - 165q^{77} - 803q^{79} - 1815q^{82} + 813q^{83} - 33q^{86} - 1221q^{88} + 396q^{89} - 781q^{91} - 858q^{92} + 2100q^{94} - 736q^{97} - 423q^{98} + O(q^{100}) \)

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
−2.37228
3.37228
−1.37228 0 −6.11684 0 0 5.11684 19.3723 0 0
1.2 4.37228 0 11.1168 0 0 −12.1168 13.6277 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 2025.4.a.n 2
3.b odd 2 1 2025.4.a.g 2
5.b even 2 1 81.4.a.a 2
9.d odd 6 2 225.4.e.b 4
15.d odd 2 1 81.4.a.d 2
20.d odd 2 1 1296.4.a.i 2
45.h odd 6 2 9.4.c.a 4
45.j even 6 2 27.4.c.a 4
45.l even 12 4 225.4.k.b 8
60.h even 2 1 1296.4.a.u 2
180.n even 6 2 144.4.i.c 4
180.p odd 6 2 432.4.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 45.h odd 6 2
27.4.c.a 4 45.j even 6 2
81.4.a.a 2 5.b even 2 1
81.4.a.d 2 15.d odd 2 1
144.4.i.c 4 180.n even 6 2
225.4.e.b 4 9.d odd 6 2
225.4.k.b 8 45.l even 12 4
432.4.i.c 4 180.p odd 6 2
1296.4.a.i 2 20.d odd 2 1
1296.4.a.u 2 60.h even 2 1
2025.4.a.g 2 3.b odd 2 1
2025.4.a.n 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2025))\):

\( T_{2}^{2} - 3 T_{2} - 6 \)
\( T_{7}^{2} + 7 T_{7} - 62 \)
\( T_{11}^{2} + 66 T_{11} + 561 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -62 + 7 T + T^{2} \)
$11$ \( 561 + 66 T + T^{2} \)
$13$ \( -1826 - 11 T + T^{2} \)
$17$ \( 1782 - 99 T + T^{2} \)
$19$ \( -4532 + 77 T + T^{2} \)
$23$ \( -2706 - 33 T + T^{2} \)
$29$ \( 642 - 51 T + T^{2} \)
$31$ \( 388 - 43 T + T^{2} \)
$37$ \( -23432 - 50 T + T^{2} \)
$41$ \( -74877 + 132 T + T^{2} \)
$43$ \( 1639 + 88 T + T^{2} \)
$47$ \( -28518 - 399 T + T^{2} \)
$53$ \( -215784 - 54 T + T^{2} \)
$59$ \( 6609 + 798 T + T^{2} \)
$61$ \( -42776 - 439 T + T^{2} \)
$67$ \( 208099 + 988 T + T^{2} \)
$71$ \( 296784 + 1368 T + T^{2} \)
$73$ \( -435398 - 455 T + T^{2} \)
$79$ \( -626516 + 803 T + T^{2} \)
$83$ \( 70788 - 813 T + T^{2} \)
$89$ \( -3564 - 396 T + T^{2} \)
$97$ \( 49591 + 736 T + T^{2} \)
show more
show less