Properties

Label 225.6.a.s
Level $225$
Weight $6$
Character orbit 225.a
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{241})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{2} + ( - 5 \beta + 37) q^{4} + ( - 4 \beta - 98) q^{7} + ( - 15 \beta + 315) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 3) q^{2} + ( - 5 \beta + 37) q^{4} + ( - 4 \beta - 98) q^{7} + ( - 15 \beta + 315) q^{8} + ( - 50 \beta + 123) q^{11} + ( - 32 \beta + 196) q^{13} + (90 \beta - 54) q^{14} + ( - 185 \beta + 661) q^{16} + ( - 136 \beta + 813) q^{17} + ( - 70 \beta - 1555) q^{19} + ( - 223 \beta + 3369) q^{22} + (12 \beta + 774) q^{23} + ( - 260 \beta + 2508) q^{26} + (362 \beta - 2426) q^{28} + (80 \beta + 1920) q^{29} + ( - 1100 \beta + 2) q^{31} + ( - 551 \beta + 3003) q^{32} + ( - 1085 \beta + 10599) q^{34} + ( - 384 \beta - 818) q^{37} + (1415 \beta - 465) q^{38} + ( - 400 \beta - 13677) q^{41} + (2128 \beta + 436) q^{43} + ( - 2215 \beta + 19551) q^{44} + ( - 750 \beta + 1602) q^{46} + (1544 \beta + 12108) q^{47} + (800 \beta - 6243) q^{49} + ( - 2004 \beta + 16852) q^{52} + (752 \beta - 13866) q^{53} + (270 \beta - 27270) q^{56} + ( - 1760 \beta + 960) q^{58} + (1960 \beta - 6960) q^{59} + (2000 \beta - 13198) q^{61} + ( - 2202 \beta + 66006) q^{62} + (1815 \beta + 20917) q^{64} + (1586 \beta + 19237) q^{67} + ( - 8417 \beta + 70881) q^{68} + ( - 1000 \beta + 44148) q^{71} + ( - 1112 \beta + 35701) q^{73} + (50 \beta + 20586) q^{74} + (5535 \beta - 36535) q^{76} + (4608 \beta - 54) q^{77} + (5020 \beta + 30230) q^{79} + (12877 \beta - 17031) q^{82} + ( - 858 \beta + 46719) q^{83} + (3820 \beta - 126372) q^{86} + ( - 16845 \beta + 83745) q^{88} + (10440 \beta + 31185) q^{89} + (2480 \beta - 11528) q^{91} + ( - 3486 \beta + 25038) q^{92} + ( - 9020 \beta - 56316) q^{94} + ( - 10944 \beta + 68542) q^{97} + (7843 \beta - 66729) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 69 q^{4} - 200 q^{7} + 615 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} + 69 q^{4} - 200 q^{7} + 615 q^{8} + 196 q^{11} + 360 q^{13} - 18 q^{14} + 1137 q^{16} + 1490 q^{17} - 3180 q^{19} + 6515 q^{22} + 1560 q^{23} + 4756 q^{26} - 4490 q^{28} + 3920 q^{29} - 1096 q^{31} + 5455 q^{32} + 20113 q^{34} - 2020 q^{37} + 485 q^{38} - 27754 q^{41} + 3000 q^{43} + 36887 q^{44} + 2454 q^{46} + 25760 q^{47} - 11686 q^{49} + 31700 q^{52} - 26980 q^{53} - 54270 q^{56} + 160 q^{58} - 11960 q^{59} - 24396 q^{61} + 129810 q^{62} + 43649 q^{64} + 40060 q^{67} + 133345 q^{68} + 87296 q^{71} + 70290 q^{73} + 41222 q^{74} - 67535 q^{76} + 4500 q^{77} + 65480 q^{79} - 21185 q^{82} + 92580 q^{83} - 248924 q^{86} + 150645 q^{88} + 72810 q^{89} - 20576 q^{91} + 46590 q^{92} - 121652 q^{94} + 126140 q^{97} - 125615 q^{98}+O(q^{100}) \) 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
8.26209
−7.26209
−5.26209 0 −4.31044 0 0 −131.048 191.069 0 0
1.2 10.2621 0 73.3104 0 0 −68.9517 423.931 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.s 2
3.b odd 2 1 25.6.a.b 2
5.b even 2 1 225.6.a.l 2
5.c odd 4 2 225.6.b.i 4
12.b even 2 1 400.6.a.w 2
15.d odd 2 1 25.6.a.d yes 2
15.e even 4 2 25.6.b.b 4
60.h even 2 1 400.6.a.o 2
60.l odd 4 2 400.6.c.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 3.b odd 2 1
25.6.a.d yes 2 15.d odd 2 1
25.6.b.b 4 15.e even 4 2
225.6.a.l 2 5.b even 2 1
225.6.a.s 2 1.a even 1 1 trivial
225.6.b.i 4 5.c odd 4 2
400.6.a.o 2 60.h even 2 1
400.6.a.w 2 12.b even 2 1
400.6.c.n 4 60.l odd 4 2

Hecke kernels

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

\( T_{2}^{2} - 5T_{2} - 54 \) Copy content Toggle raw display
\( T_{7}^{2} + 200T_{7} + 9036 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5T - 54 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 200T + 9036 \) Copy content Toggle raw display
$11$ \( T^{2} - 196T - 141021 \) Copy content Toggle raw display
$13$ \( T^{2} - 360T - 29296 \) Copy content Toggle raw display
$17$ \( T^{2} - 1490 T - 559359 \) Copy content Toggle raw display
$19$ \( T^{2} + 3180 T + 2232875 \) Copy content Toggle raw display
$23$ \( T^{2} - 1560 T + 599724 \) Copy content Toggle raw display
$29$ \( T^{2} - 3920 T + 3456000 \) Copy content Toggle raw display
$31$ \( T^{2} + 1096 T - 72602196 \) Copy content Toggle raw display
$37$ \( T^{2} + 2020 T - 7864124 \) Copy content Toggle raw display
$41$ \( T^{2} + 27754 T + 182931129 \) Copy content Toggle raw display
$43$ \( T^{2} - 3000 T - 270585136 \) Copy content Toggle raw display
$47$ \( T^{2} - 25760 T + 22262256 \) Copy content Toggle raw display
$53$ \( T^{2} + 26980 T + 147908484 \) Copy content Toggle raw display
$59$ \( T^{2} + 11960 T - 195696000 \) Copy content Toggle raw display
$61$ \( T^{2} + 24396 T - 92208796 \) Copy content Toggle raw display
$67$ \( T^{2} - 40060 T + 249648291 \) Copy content Toggle raw display
$71$ \( T^{2} - 87296 T + 1844897904 \) Copy content Toggle raw display
$73$ \( T^{2} - 70290 T + 1160669249 \) Copy content Toggle raw display
$79$ \( T^{2} - 65480 T - 446416500 \) Copy content Toggle raw display
$83$ \( T^{2} - 92580 T + 2098410219 \) Copy content Toggle raw display
$89$ \( T^{2} - 72810 T - 5241540375 \) Copy content Toggle raw display
$97$ \( T^{2} - 126140 T - 3238386044 \) Copy content Toggle raw display
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