Properties

Label 225.6.a.u
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.

\(f(q)\) \(=\) \( q + ( - \beta + 5) q^{2} + ( - 9 \beta + 15) q^{4} + (24 \beta - 66) q^{7} + ( - 19 \beta + 113) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 5) q^{2} + ( - 9 \beta + 15) q^{4} + (24 \beta - 66) q^{7} + ( - 19 \beta + 113) q^{8} + (108 \beta - 138) q^{11} + ( - 84 \beta - 606) q^{13} + (162 \beta - 858) q^{14} + (99 \beta + 503) q^{16} + (140 \beta + 218) q^{17} + (72 \beta - 704) q^{19} + (570 \beta - 3066) q^{22} + (88 \beta + 2908) q^{23} + (270 \beta - 1182) q^{26} + (738 \beta - 5742) q^{28} + ( - 864 \beta + 2208) q^{29} + (144 \beta - 5896) q^{31} + (501 \beta - 3279) q^{32} + (342 \beta - 1990) q^{34} + ( - 396 \beta - 7146) q^{37} + (992 \beta - 5104) q^{38} + ( - 3024 \beta + 606) q^{41} + (1344 \beta + 3108) q^{43} + (1890 \beta - 23454) q^{44} + ( - 2556 \beta + 12604) q^{46} + ( - 1288 \beta + 2264) q^{47} + ( - 2592 \beta + 221) q^{49} + (4950 \beta + 7542) q^{52} + ( - 2012 \beta - 10082) q^{53} + (3510 \beta - 17490) q^{56} + ( - 5664 \beta + 30048) q^{58} + ( - 3348 \beta - 26994) q^{59} + (3168 \beta - 16654) q^{61} + (6472 \beta - 32648) q^{62} + (2115 \beta - 43513) q^{64} + ( - 4560 \beta + 4872) q^{67} + ( - 1122 \beta - 24450) q^{68} + (3240 \beta - 10332) q^{71} + (792 \beta + 1332) q^{73} + (5562 \beta - 27018) q^{74} + (6768 \beta - 24816) q^{76} + ( - 7848 \beta + 66132) q^{77} + (9360 \beta - 33440) q^{79} + ( - 12702 \beta + 69558) q^{82} + (6832 \beta + 63580) q^{83} + (2268 \beta - 14028) q^{86} + (12774 \beta - 60738) q^{88} + ( - 4752 \beta - 66006) q^{89} + ( - 11016 \beta - 4356) q^{91} + ( - 25644 \beta + 26196) q^{92} + ( - 7416 \beta + 39656) q^{94} + ( - 15600 \beta - 34968) q^{97} + ( - 10589 \beta + 58129) q^{98}+O(q^{100}) \) 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
−0.216991 0 −31.9529 0 0 59.2078 13.8772 0 0
1.2 9.21699 0 52.9529 0 0 −167.208 193.123 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.u 2
3.b odd 2 1 75.6.a.f 2
5.b even 2 1 225.6.a.i 2
5.c odd 4 2 45.6.b.c 4
15.d odd 2 1 75.6.a.j 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 3.b odd 2 1
75.6.a.j 2 15.d odd 2 1
225.6.a.i 2 5.b even 2 1
225.6.a.u 2 1.a even 1 1 trivial
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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