Properties

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

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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: \(\Q(\sqrt{70}) \)
Defining polynomial: \( x^{2} - 70 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{70}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 38 q^{4} - 45 q^{7} + 6 \beta q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 38 q^{4} - 45 q^{7} + 6 \beta q^{8} - 40 \beta q^{11} - 1045 q^{13} - 45 \beta q^{14} - 796 q^{16} - 152 \beta q^{17} + 1159 q^{19} - 2800 q^{22} + 456 \beta q^{23} - 1045 \beta q^{26} - 1710 q^{28} - 440 \beta q^{29} + 3633 q^{31} - 988 \beta q^{32} - 10640 q^{34} - 3110 q^{37} + 1159 \beta q^{38} - 2120 \beta q^{41} + 10355 q^{43} - 1520 \beta q^{44} + 31920 q^{46} - 1616 \beta q^{47} - 14782 q^{49} - 39710 q^{52} + 1192 \beta q^{53} - 270 \beta q^{56} - 30800 q^{58} + 3440 \beta q^{59} - 7613 q^{61} + 3633 \beta q^{62} - 43688 q^{64} - 50445 q^{67} - 5776 \beta q^{68} + 9640 \beta q^{71} - 74710 q^{73} - 3110 \beta q^{74} + 44042 q^{76} + 1800 \beta q^{77} + 38316 q^{79} - 148400 q^{82} - 4272 \beta q^{83} + 10355 \beta q^{86} - 16800 q^{88} + 14400 \beta q^{89} + 47025 q^{91} + 17328 \beta q^{92} - 113120 q^{94} + 71755 q^{97} - 14782 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 76 q^{4} - 90 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 76 q^{4} - 90 q^{7} - 2090 q^{13} - 1592 q^{16} + 2318 q^{19} - 5600 q^{22} - 3420 q^{28} + 7266 q^{31} - 21280 q^{34} - 6220 q^{37} + 20710 q^{43} + 63840 q^{46} - 29564 q^{49} - 79420 q^{52} - 61600 q^{58} - 15226 q^{61} - 87376 q^{64} - 100890 q^{67} - 149420 q^{73} + 88084 q^{76} + 76632 q^{79} - 296800 q^{82} - 33600 q^{88} + 94050 q^{91} - 226240 q^{94} + 143510 q^{97}+O(q^{100}) \) 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
−8.36660
8.36660
−8.36660 0 38.0000 0 0 −45.0000 −50.1996 0 0
1.2 8.36660 0 38.0000 0 0 −45.0000 50.1996 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.6.a.o 2
3.b odd 2 1 inner 225.6.a.o 2
5.b even 2 1 225.6.a.p yes 2
5.c odd 4 2 225.6.b.h 4
15.d odd 2 1 225.6.a.p yes 2
15.e even 4 2 225.6.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.6.a.o 2 1.a even 1 1 trivial
225.6.a.o 2 3.b odd 2 1 inner
225.6.a.p yes 2 5.b even 2 1
225.6.a.p yes 2 15.d odd 2 1
225.6.b.h 4 5.c odd 4 2
225.6.b.h 4 15.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 70 \) Copy content Toggle raw display
\( T_{7} + 45 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 70 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 45)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 112000 \) Copy content Toggle raw display
$13$ \( (T + 1045)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1617280 \) Copy content Toggle raw display
$19$ \( (T - 1159)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 14555520 \) Copy content Toggle raw display
$29$ \( T^{2} - 13552000 \) Copy content Toggle raw display
$31$ \( (T - 3633)^{2} \) Copy content Toggle raw display
$37$ \( (T + 3110)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 314608000 \) Copy content Toggle raw display
$43$ \( (T - 10355)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 182801920 \) Copy content Toggle raw display
$53$ \( T^{2} - 99460480 \) Copy content Toggle raw display
$59$ \( T^{2} - 828352000 \) Copy content Toggle raw display
$61$ \( (T + 7613)^{2} \) Copy content Toggle raw display
$67$ \( (T + 50445)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 6505072000 \) Copy content Toggle raw display
$73$ \( (T + 74710)^{2} \) Copy content Toggle raw display
$79$ \( (T - 38316)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 1277498880 \) Copy content Toggle raw display
$89$ \( T^{2} - 14515200000 \) Copy content Toggle raw display
$97$ \( (T - 71755)^{2} \) Copy content Toggle raw display
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