Properties

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

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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: \(\Q(\sqrt{89}) \)
Defining polynomial: \( x^{2} - x - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{89})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 4) q^{2} + (9 \beta + 6) q^{4} + (24 \beta + 42) q^{7} + ( - 19 \beta - 94) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 4) q^{2} + (9 \beta + 6) q^{4} + (24 \beta + 42) q^{7} + ( - 19 \beta - 94) q^{8} + ( - 108 \beta - 30) q^{11} + ( - 84 \beta + 690) q^{13} + ( - 162 \beta - 696) q^{14} + ( - 99 \beta + 602) q^{16} + (140 \beta - 358) q^{17} + ( - 72 \beta - 632) q^{19} + (570 \beta + 2496) q^{22} + (88 \beta - 2996) q^{23} + ( - 270 \beta - 912) q^{26} + (738 \beta + 5004) q^{28} + (864 \beta + 1344) q^{29} + ( - 144 \beta - 5752) q^{31} + (501 \beta + 2778) q^{32} + ( - 342 \beta - 1648) q^{34} + ( - 396 \beta + 7542) q^{37} + (992 \beta + 4112) q^{38} + (3024 \beta - 2418) q^{41} + (1344 \beta - 4452) q^{43} + ( - 1890 \beta - 21564) q^{44} + (2556 \beta + 10048) q^{46} + ( - 1288 \beta - 976) q^{47} + (2592 \beta - 2371) q^{49} + (4950 \beta - 12492) q^{52} + ( - 2012 \beta + 12094) q^{53} + ( - 3510 \beta - 13980) q^{56} + ( - 5664 \beta - 24384) q^{58} + (3348 \beta - 30342) q^{59} + ( - 3168 \beta - 13486) q^{61} + (6472 \beta + 26176) q^{62} + ( - 2115 \beta - 41398) q^{64} + ( - 4560 \beta - 312) q^{67} + ( - 1122 \beta + 25572) q^{68} + ( - 3240 \beta - 7092) q^{71} + (792 \beta - 2124) q^{73} + ( - 5562 \beta - 21456) q^{74} + ( - 6768 \beta - 18048) q^{76} + ( - 7848 \beta - 58284) q^{77} + ( - 9360 \beta - 24080) q^{79} + ( - 12702 \beta - 56856) q^{82} + (6832 \beta - 70412) q^{83} + ( - 2268 \beta - 11760) q^{86} + (12774 \beta + 47964) q^{88} + (4752 \beta - 70758) q^{89} + (11016 \beta - 15372) q^{91} + ( - 25644 \beta - 552) q^{92} + (7416 \beta + 32240) q^{94} + ( - 15600 \beta + 50568) q^{97} + ( - 10589 \beta - 47540) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8} - 168 q^{11} + 1296 q^{13} - 1554 q^{14} + 1105 q^{16} - 576 q^{17} - 1336 q^{19} + 5562 q^{22} - 5904 q^{23} - 2094 q^{26} + 10746 q^{28} + 3552 q^{29} - 11648 q^{31} + 6057 q^{32} - 3638 q^{34} + 14688 q^{37} + 9216 q^{38} - 1812 q^{41} - 7560 q^{43} - 45018 q^{44} + 22652 q^{46} - 3240 q^{47} - 2150 q^{49} - 20034 q^{52} + 22176 q^{53} - 31470 q^{56} - 54432 q^{58} - 57336 q^{59} - 30140 q^{61} + 58824 q^{62} - 84911 q^{64} - 5184 q^{67} + 50022 q^{68} - 17424 q^{71} - 3456 q^{73} - 48474 q^{74} - 42864 q^{76} - 124416 q^{77} - 57520 q^{79} - 126414 q^{82} - 133992 q^{83} - 25788 q^{86} + 108702 q^{88} - 136764 q^{89} - 19728 q^{91} - 26748 q^{92} + 71896 q^{94} + 85536 q^{97} - 105669 q^{98}+O(q^{100}) \) 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
5.21699
−4.21699
−9.21699 0 52.9529 0 0 167.208 −193.123 0 0
1.2 0.216991 0 −31.9529 0 0 −59.2078 −13.8772 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.i 2
3.b odd 2 1 75.6.a.j 2
5.b even 2 1 225.6.a.u 2
5.c odd 4 2 45.6.b.c 4
15.d odd 2 1 75.6.a.f 2
15.e even 4 2 15.6.b.a 4
20.e even 4 2 720.6.f.h 4
60.l odd 4 2 240.6.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.b.a 4 15.e even 4 2
45.6.b.c 4 5.c odd 4 2
75.6.a.f 2 15.d odd 2 1
75.6.a.j 2 3.b odd 2 1
225.6.a.i 2 1.a even 1 1 trivial
225.6.a.u 2 5.b even 2 1
240.6.f.c 4 60.l odd 4 2
720.6.f.h 4 20.e even 4 2

Hecke kernels

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

\( T_{2}^{2} + 9T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 108T_{7} - 9900 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 108T - 9900 \) Copy content Toggle raw display
$11$ \( T^{2} + 168T - 252468 \) Copy content Toggle raw display
$13$ \( T^{2} - 1296 T + 262908 \) Copy content Toggle raw display
$17$ \( T^{2} + 576T - 353156 \) Copy content Toggle raw display
$19$ \( T^{2} + 1336 T + 330880 \) Copy content Toggle raw display
$23$ \( T^{2} + 5904 T + 8542000 \) Copy content Toggle raw display
$29$ \( T^{2} - 3552 T - 13455360 \) Copy content Toggle raw display
$31$ \( T^{2} + 11648 T + 33457600 \) Copy content Toggle raw display
$37$ \( T^{2} - 14688 T + 50445180 \) Copy content Toggle raw display
$41$ \( T^{2} + 1812 T - 202645980 \) Copy content Toggle raw display
$43$ \( T^{2} + 7560 T - 25902576 \) Copy content Toggle raw display
$47$ \( T^{2} + 3240 T - 34287104 \) Copy content Toggle raw display
$53$ \( T^{2} - 22176 T + 32872540 \) Copy content Toggle raw display
$59$ \( T^{2} + 57336 T + 572451660 \) Copy content Toggle raw display
$61$ \( T^{2} + 30140 T + 3798916 \) Copy content Toggle raw display
$67$ \( T^{2} + 5184 T - 455939136 \) Copy content Toggle raw display
$71$ \( T^{2} + 17424 T - 157672656 \) Copy content Toggle raw display
$73$ \( T^{2} + 3456 T - 10970640 \) Copy content Toggle raw display
$79$ \( T^{2} + 57520 T - 1122176000 \) Copy content Toggle raw display
$83$ \( T^{2} + 133992 T + 3449918032 \) Copy content Toggle raw display
$89$ \( T^{2} + 136764 T + 4173659460 \) Copy content Toggle raw display
$97$ \( T^{2} - 85536 T - 3585658176 \) Copy content Toggle raw display
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