Properties

Label 225.6.a.l
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: \(\Q(\sqrt{241}) \)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 2) q^{2} + (5 \beta + 32) q^{4} + ( - 4 \beta + 102) q^{7} + ( - 15 \beta - 300) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 2) q^{2} + (5 \beta + 32) q^{4} + ( - 4 \beta + 102) q^{7} + ( - 15 \beta - 300) q^{8} + (50 \beta + 73) q^{11} + ( - 32 \beta - 164) q^{13} + ( - 90 \beta + 36) q^{14} + (185 \beta + 476) q^{16} + ( - 136 \beta - 677) q^{17} + (70 \beta - 1625) q^{19} + ( - 223 \beta - 3146) q^{22} + (12 \beta - 786) q^{23} + (260 \beta + 2248) q^{26} + (362 \beta + 2064) q^{28} + ( - 80 \beta + 2000) q^{29} + (1100 \beta - 1098) q^{31} + ( - 551 \beta - 2452) q^{32} + (1085 \beta + 9514) q^{34} + ( - 384 \beta + 1202) q^{37} + (1415 \beta - 950) q^{38} + (400 \beta - 14077) q^{41} + (2128 \beta - 2564) q^{43} + (2215 \beta + 17336) q^{44} + (750 \beta + 852) q^{46} + (1544 \beta - 13652) q^{47} + ( - 800 \beta - 5443) q^{49} + ( - 2004 \beta - 14848) q^{52} + (752 \beta + 13114) q^{53} + ( - 270 \beta - 27000) q^{56} + ( - 1760 \beta + 800) q^{58} + ( - 1960 \beta - 5000) q^{59} + ( - 2000 \beta - 11198) q^{61} + ( - 2202 \beta - 63804) q^{62} + ( - 1815 \beta + 22732) q^{64} + (1586 \beta - 20823) q^{67} + ( - 8417 \beta - 62464) q^{68} + (1000 \beta + 43148) q^{71} + ( - 1112 \beta - 34589) q^{73} + ( - 50 \beta + 20636) q^{74} + ( - 5535 \beta - 31000) q^{76} + (4608 \beta - 4554) q^{77} + ( - 5020 \beta + 35250) q^{79} + (12877 \beta + 4154) q^{82} + ( - 858 \beta - 45861) q^{83} + ( - 3820 \beta - 122552) q^{86} + ( - 16845 \beta - 66900) q^{88} + ( - 10440 \beta + 41625) q^{89} + ( - 2480 \beta - 9048) q^{91} + ( - 3486 \beta - 21552) q^{92} + (9020 \beta - 65336) q^{94} + ( - 10944 \beta - 57598) q^{97} + (7843 \beta + 58886) q^{98}+O(q^{100}) \) 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
−10.2621 0 73.3104 0 0 68.9517 −423.931 0 0
1.2 5.26209 0 −4.31044 0 0 131.048 −191.069 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.l 2
3.b odd 2 1 25.6.a.d yes 2
5.b even 2 1 225.6.a.s 2
5.c odd 4 2 225.6.b.i 4
12.b even 2 1 400.6.a.o 2
15.d odd 2 1 25.6.a.b 2
15.e even 4 2 25.6.b.b 4
60.h even 2 1 400.6.a.w 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 15.d odd 2 1
25.6.a.d yes 2 3.b odd 2 1
25.6.b.b 4 15.e even 4 2
225.6.a.l 2 1.a even 1 1 trivial
225.6.a.s 2 5.b even 2 1
225.6.b.i 4 5.c odd 4 2
400.6.a.o 2 12.b even 2 1
400.6.a.w 2 60.h 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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