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

Related objects

Downloads

Learn more about

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: \(x^{2} - 11\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + 12 q^{4} + 9 \beta q^{7} -20 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 12 q^{4} + 9 \beta q^{7} -20 \beta q^{8} -252 q^{11} -18 \beta q^{13} + 396 q^{14} -1264 q^{16} -104 \beta q^{17} + 220 q^{19} -252 \beta q^{22} -367 \beta q^{23} -792 q^{26} + 108 \beta q^{28} -6930 q^{29} + 6752 q^{31} -624 \beta q^{32} -4576 q^{34} -2106 \beta q^{37} + 220 \beta q^{38} + 198 q^{41} -63 \beta q^{43} -3024 q^{44} -16148 q^{46} -1589 \beta q^{47} -13243 q^{49} -216 \beta q^{52} + 878 \beta q^{53} -7920 q^{56} -6930 \beta q^{58} -24660 q^{59} -5698 q^{61} + 6752 \beta q^{62} + 12992 q^{64} + 6579 \beta q^{67} -1248 \beta q^{68} -53352 q^{71} + 10692 \beta q^{73} -92664 q^{74} + 2640 q^{76} -2268 \beta q^{77} -51920 q^{79} + 198 \beta q^{82} + 9323 \beta q^{83} -2772 q^{86} + 5040 \beta q^{88} -9990 q^{89} -7128 q^{91} -4404 \beta q^{92} -69916 q^{94} + 15264 \beta q^{97} -13243 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 24q^{4} + O(q^{10}) \) \( 2q + 24q^{4} - 504q^{11} + 792q^{14} - 2528q^{16} + 440q^{19} - 1584q^{26} - 13860q^{29} + 13504q^{31} - 9152q^{34} + 396q^{41} - 6048q^{44} - 32296q^{46} - 26486q^{49} - 15840q^{56} - 49320q^{59} - 11396q^{61} + 25984q^{64} - 106704q^{71} - 185328q^{74} + 5280q^{76} - 103840q^{79} - 5544q^{86} - 19980q^{89} - 14256q^{91} - 139832q^{94} + O(q^{100}) \)

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
−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:

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

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{2}^{2} - 44 \)
\( T_{7}^{2} - 3564 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 20 T^{2} + 1024 T^{4} \)
$3$ 1
$5$ 1
$7$ \( 1 + 30050 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 + 252 T + 161051 T^{2} )^{2} \)
$13$ \( 1 + 728330 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 + 2363810 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 - 220 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 + 6946370 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 6930 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 - 6752 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 56462470 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 - 198 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 + 293842250 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 + 347593490 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 + 802472090 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 24660 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 + 5698 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 + 795787610 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 53352 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 883886830 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 + 51920 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 + 4053674810 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 9990 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 + 6923133890 T^{2} + 73742412689492826049 T^{4} \)
show more
show less