Properties

Label 225.6.a.n
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: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 12 q^{4} + 9 \beta q^{7} - 20 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 12 q^{4} + 9 \beta q^{7} - 20 \beta q^{8} - 252 q^{11} - 18 \beta q^{13} + 396 q^{14} - 1264 q^{16} - 104 \beta q^{17} + 220 q^{19} - 252 \beta q^{22} - 367 \beta q^{23} - 792 q^{26} + 108 \beta q^{28} - 6930 q^{29} + 6752 q^{31} - 624 \beta q^{32} - 4576 q^{34} - 2106 \beta q^{37} + 220 \beta q^{38} + 198 q^{41} - 63 \beta q^{43} - 3024 q^{44} - 16148 q^{46} - 1589 \beta q^{47} - 13243 q^{49} - 216 \beta q^{52} + 878 \beta q^{53} - 7920 q^{56} - 6930 \beta q^{58} - 24660 q^{59} - 5698 q^{61} + 6752 \beta q^{62} + 12992 q^{64} + 6579 \beta q^{67} - 1248 \beta q^{68} - 53352 q^{71} + 10692 \beta q^{73} - 92664 q^{74} + 2640 q^{76} - 2268 \beta q^{77} - 51920 q^{79} + 198 \beta q^{82} + 9323 \beta q^{83} - 2772 q^{86} + 5040 \beta q^{88} - 9990 q^{89} - 7128 q^{91} - 4404 \beta q^{92} - 69916 q^{94} + 15264 \beta q^{97} - 13243 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{4} - 504 q^{11} + 792 q^{14} - 2528 q^{16} + 440 q^{19} - 1584 q^{26} - 13860 q^{29} + 13504 q^{31} - 9152 q^{34} + 396 q^{41} - 6048 q^{44} - 32296 q^{46} - 26486 q^{49} - 15840 q^{56} - 49320 q^{59} - 11396 q^{61} + 25984 q^{64} - 106704 q^{71} - 185328 q^{74} + 5280 q^{76} - 103840 q^{79} - 5544 q^{86} - 19980 q^{89} - 14256 q^{91} - 139832 q^{94}+O(q^{100}) \) 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
−3.31662
3.31662
−6.63325 0 12.0000 0 0 −59.6992 132.665 0 0
1.2 6.63325 0 12.0000 0 0 59.6992 −132.665 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
5.b even 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.n 2
3.b odd 2 1 25.6.a.c 2
5.b even 2 1 inner 225.6.a.n 2
5.c odd 4 2 45.6.b.b 2
12.b even 2 1 400.6.a.t 2
15.d odd 2 1 25.6.a.c 2
15.e even 4 2 5.6.b.a 2
20.e even 4 2 720.6.f.f 2
60.h even 2 1 400.6.a.t 2
60.l odd 4 2 80.6.c.a 2
105.k odd 4 2 245.6.b.a 2
120.q odd 4 2 320.6.c.g 2
120.w even 4 2 320.6.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 15.e even 4 2
25.6.a.c 2 3.b odd 2 1
25.6.a.c 2 15.d odd 2 1
45.6.b.b 2 5.c odd 4 2
80.6.c.a 2 60.l odd 4 2
225.6.a.n 2 1.a even 1 1 trivial
225.6.a.n 2 5.b even 2 1 inner
245.6.b.a 2 105.k odd 4 2
320.6.c.f 2 120.w even 4 2
320.6.c.g 2 120.q odd 4 2
400.6.a.t 2 12.b even 2 1
400.6.a.t 2 60.h even 2 1
720.6.f.f 2 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 44 \) Copy content Toggle raw display
\( T_{7}^{2} - 3564 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 44 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3564 \) Copy content Toggle raw display
$11$ \( (T + 252)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 14256 \) Copy content Toggle raw display
$17$ \( T^{2} - 475904 \) Copy content Toggle raw display
$19$ \( (T - 220)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 5926316 \) Copy content Toggle raw display
$29$ \( (T + 6930)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6752)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 195150384 \) Copy content Toggle raw display
$41$ \( (T - 198)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 174636 \) Copy content Toggle raw display
$47$ \( T^{2} - 111096524 \) Copy content Toggle raw display
$53$ \( T^{2} - 33918896 \) Copy content Toggle raw display
$59$ \( (T + 24660)^{2} \) Copy content Toggle raw display
$61$ \( (T + 5698)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 1904462604 \) Copy content Toggle raw display
$71$ \( (T + 53352)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 5030030016 \) Copy content Toggle raw display
$79$ \( (T + 51920)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 3824406476 \) Copy content Toggle raw display
$89$ \( (T + 9990)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 10251546624 \) Copy content Toggle raw display
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