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: \(x^{2} - x - 60\)
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.

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

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 \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 5 T_{2} - 54 \)
\( T_{7}^{2} + 200 T_{7} + 9036 \)

Hecke characteristic polynomials

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