Properties

Label 225.6.a.t
Level $225$
Weight $6$
Character orbit 225.a
Self dual yes
Analytic conductor $36.086$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.0863594579\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
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 \(\beta = \sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{2} + (6 \beta + 8) q^{4} + ( - 24 \beta - 51) q^{7} + ( - 6 \beta + 114) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 3) q^{2} + (6 \beta + 8) q^{4} + ( - 24 \beta - 51) q^{7} + ( - 6 \beta + 114) q^{8} + (88 \beta + 6) q^{11} + ( - 96 \beta - 527) q^{13} + ( - 123 \beta - 897) q^{14} + ( - 96 \beta - 100) q^{16} + (40 \beta - 858) q^{17} + ( - 168 \beta + 2107) q^{19} + (270 \beta + 2746) q^{22} + ( - 648 \beta - 222) q^{23} + ( - 815 \beta - 4557) q^{26} + ( - 498 \beta - 4872) q^{28} + (56 \beta - 2034) q^{29} + ( - 936 \beta - 1299) q^{31} + ( - 196 \beta - 6924) q^{32} + ( - 738 \beta - 1334) q^{34} + (1776 \beta - 2206) q^{37} + (1603 \beta + 1113) q^{38} + (296 \beta - 5616) q^{41} + (216 \beta + 4225) q^{43} + (740 \beta + 16416) q^{44} + ( - 2166 \beta - 20754) q^{46} + (3328 \beta + 1230) q^{47} + (2448 \beta + 3650) q^{49} + ( - 3930 \beta - 22072) q^{52} + ( - 248 \beta - 32532) q^{53} + ( - 2430 \beta - 1350) q^{56} + ( - 1866 \beta - 4366) q^{58} + ( - 3008 \beta + 31962) q^{59} + (1008 \beta + 3655) q^{61} + ( - 4107 \beta - 32913) q^{62} + ( - 4440 \beta - 23648) q^{64} + (5160 \beta - 30867) q^{67} + ( - 4828 \beta + 576) q^{68} + (6200 \beta - 49152) q^{71} + (3408 \beta + 13282) q^{73} + (3122 \beta + 48438) q^{74} + (11298 \beta - 14392) q^{76} + ( - 4632 \beta - 65778) q^{77} + (1920 \beta + 42000) q^{79} + ( - 4728 \beta - 7672) q^{82} + (768 \beta - 32886) q^{83} + (4873 \beta + 19371) q^{86} + (9996 \beta - 15684) q^{88} + (3168 \beta - 51552) q^{89} + (17544 \beta + 98301) q^{91} + ( - 6516 \beta - 122304) q^{92} + (11214 \beta + 106858) q^{94} + ( - 22080 \beta - 34687) q^{97} + (10994 \beta + 86838) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 16 q^{4} - 102 q^{7} + 228 q^{8} + 12 q^{11} - 1054 q^{13} - 1794 q^{14} - 200 q^{16} - 1716 q^{17} + 4214 q^{19} + 5492 q^{22} - 444 q^{23} - 9114 q^{26} - 9744 q^{28} - 4068 q^{29} - 2598 q^{31} - 13848 q^{32} - 2668 q^{34} - 4412 q^{37} + 2226 q^{38} - 11232 q^{41} + 8450 q^{43} + 32832 q^{44} - 41508 q^{46} + 2460 q^{47} + 7300 q^{49} - 44144 q^{52} - 65064 q^{53} - 2700 q^{56} - 8732 q^{58} + 63924 q^{59} + 7310 q^{61} - 65826 q^{62} - 47296 q^{64} - 61734 q^{67} + 1152 q^{68} - 98304 q^{71} + 26564 q^{73} + 96876 q^{74} - 28784 q^{76} - 131556 q^{77} + 84000 q^{79} - 15344 q^{82} - 65772 q^{83} + 38742 q^{86} - 31368 q^{88} - 103104 q^{89} + 196602 q^{91} - 244608 q^{92} + 213716 q^{94} - 69374 q^{97} + 173676 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−5.56776
5.56776
−2.56776 0 −25.4066 0 0 82.6263 147.407 0 0
1.2 8.56776 0 41.4066 0 0 −184.626 80.5934 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 225.6.a.t 2
3.b odd 2 1 75.6.a.g 2
5.b even 2 1 225.6.a.j 2
5.c odd 4 2 225.6.b.l 4
15.d odd 2 1 75.6.a.i yes 2
15.e even 4 2 75.6.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.6.a.g 2 3.b odd 2 1
75.6.a.i yes 2 15.d odd 2 1
75.6.b.f 4 15.e even 4 2
225.6.a.j 2 5.b even 2 1
225.6.a.t 2 1.a even 1 1 trivial
225.6.b.l 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 6T_{2} - 22 \) Copy content Toggle raw display
\( T_{7}^{2} + 102T_{7} - 15255 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T - 22 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 102T - 15255 \) Copy content Toggle raw display
$11$ \( T^{2} - 12T - 240028 \) Copy content Toggle raw display
$13$ \( T^{2} + 1054T - 7967 \) Copy content Toggle raw display
$17$ \( T^{2} + 1716 T + 686564 \) Copy content Toggle raw display
$19$ \( T^{2} - 4214 T + 3564505 \) Copy content Toggle raw display
$23$ \( T^{2} + 444 T - 12967740 \) Copy content Toggle raw display
$29$ \( T^{2} + 4068 T + 4039940 \) Copy content Toggle raw display
$31$ \( T^{2} + 2598 T - 25471575 \) Copy content Toggle raw display
$37$ \( T^{2} + 4412 T - 92913020 \) Copy content Toggle raw display
$41$ \( T^{2} + 11232 T + 28823360 \) Copy content Toggle raw display
$43$ \( T^{2} - 8450 T + 16404289 \) Copy content Toggle raw display
$47$ \( T^{2} - 2460 T - 341830204 \) Copy content Toggle raw display
$53$ \( T^{2} + 65064 T + 1056424400 \) Copy content Toggle raw display
$59$ \( T^{2} - 63924 T + 741079460 \) Copy content Toggle raw display
$61$ \( T^{2} - 7310 T - 18138959 \) Copy content Toggle raw display
$67$ \( T^{2} + 61734 T + 127378089 \) Copy content Toggle raw display
$71$ \( T^{2} + 98304 T + 1224279104 \) Copy content Toggle raw display
$73$ \( T^{2} - 26564 T - 183636860 \) Copy content Toggle raw display
$79$ \( T^{2} - 84000 T + 1649721600 \) Copy content Toggle raw display
$83$ \( T^{2} + 65772 T + 1063204452 \) Copy content Toggle raw display
$89$ \( T^{2} + 103104 T + 2346485760 \) Copy content Toggle raw display
$97$ \( T^{2} + 69374 T - 13910130431 \) Copy content Toggle raw display
show more
show less