Properties

Label 225.6.b.e
Level 225
Weight 6
Character orbit 225.b
Analytic conductor 36.086
Analytic rank 0
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 28 q^{4} -192 i q^{7} + 120 i q^{8} +O(q^{10})\) \( q + 2 i q^{2} + 28 q^{4} -192 i q^{7} + 120 i q^{8} + 148 q^{11} + 286 i q^{13} + 384 q^{14} + 656 q^{16} -1678 i q^{17} -1060 q^{19} + 296 i q^{22} -2976 i q^{23} -572 q^{26} -5376 i q^{28} -3410 q^{29} -2448 q^{31} + 5152 i q^{32} + 3356 q^{34} -182 i q^{37} -2120 i q^{38} + 9398 q^{41} -1244 i q^{43} + 4144 q^{44} + 5952 q^{46} -12088 i q^{47} -20057 q^{49} + 8008 i q^{52} -23846 i q^{53} + 23040 q^{56} -6820 i q^{58} -20020 q^{59} + 32302 q^{61} -4896 i q^{62} + 10688 q^{64} -60972 i q^{67} -46984 i q^{68} + 32648 q^{71} -38774 i q^{73} + 364 q^{74} -29680 q^{76} -28416 i q^{77} + 33360 q^{79} + 18796 i q^{82} -16716 i q^{83} + 2488 q^{86} + 17760 i q^{88} + 101370 q^{89} + 54912 q^{91} -83328 i q^{92} + 24176 q^{94} + 119038 i q^{97} -40114 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 56q^{4} + O(q^{10}) \) \( 2q + 56q^{4} + 296q^{11} + 768q^{14} + 1312q^{16} - 2120q^{19} - 1144q^{26} - 6820q^{29} - 4896q^{31} + 6712q^{34} + 18796q^{41} + 8288q^{44} + 11904q^{46} - 40114q^{49} + 46080q^{56} - 40040q^{59} + 64604q^{61} + 21376q^{64} + 65296q^{71} + 728q^{74} - 59360q^{76} + 66720q^{79} + 4976q^{86} + 202740q^{89} + 109824q^{91} + 48352q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
2.00000i 0 28.0000 0 0 192.000i 120.000i 0 0
199.2 2.00000i 0 28.0000 0 0 192.000i 120.000i 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.b.e 2
3.b odd 2 1 25.6.b.a 2
5.b even 2 1 inner 225.6.b.e 2
5.c odd 4 1 45.6.a.b 1
5.c odd 4 1 225.6.a.f 1
12.b even 2 1 400.6.c.j 2
15.d odd 2 1 25.6.b.a 2
15.e even 4 1 5.6.a.a 1
15.e even 4 1 25.6.a.a 1
20.e even 4 1 720.6.a.a 1
60.h even 2 1 400.6.c.j 2
60.l odd 4 1 80.6.a.e 1
60.l odd 4 1 400.6.a.g 1
105.k odd 4 1 245.6.a.b 1
120.q odd 4 1 320.6.a.g 1
120.w even 4 1 320.6.a.j 1
165.l odd 4 1 605.6.a.a 1
195.s even 4 1 845.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 15.e even 4 1
25.6.a.a 1 15.e even 4 1
25.6.b.a 2 3.b odd 2 1
25.6.b.a 2 15.d odd 2 1
45.6.a.b 1 5.c odd 4 1
80.6.a.e 1 60.l odd 4 1
225.6.a.f 1 5.c odd 4 1
225.6.b.e 2 1.a even 1 1 trivial
225.6.b.e 2 5.b even 2 1 inner
245.6.a.b 1 105.k odd 4 1
320.6.a.g 1 120.q odd 4 1
320.6.a.j 1 120.w even 4 1
400.6.a.g 1 60.l odd 4 1
400.6.c.j 2 12.b even 2 1
400.6.c.j 2 60.h even 2 1
605.6.a.a 1 165.l odd 4 1
720.6.a.a 1 20.e even 4 1
845.6.a.b 1 195.s even 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \)
\( T_{7}^{2} + 36864 \)
\( T_{11} - 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 60 T^{2} + 1024 T^{4} \)
$3$ 1
$5$ 1
$7$ \( 1 + 3250 T^{2} + 282475249 T^{4} \)
$11$ \( ( 1 - 148 T + 161051 T^{2} )^{2} \)
$13$ \( 1 - 660790 T^{2} + 137858491849 T^{4} \)
$17$ \( 1 - 24030 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 1060 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 4016110 T^{2} + 41426511213649 T^{4} \)
$29$ \( ( 1 + 3410 T + 20511149 T^{2} )^{2} \)
$31$ \( ( 1 + 2448 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 138654790 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 - 9398 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 292469350 T^{2} + 21611482313284249 T^{4} \)
$47$ \( 1 - 312570270 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 267759270 T^{2} + 174887470365513049 T^{4} \)
$59$ \( ( 1 + 20020 T + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 - 32302 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 + 1017334570 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 32648 T + 1804229351 T^{2} )^{2} \)
$73$ \( 1 - 2642720110 T^{2} + 4297625829703557649 T^{4} \)
$79$ \( ( 1 - 33360 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 7598656630 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 101370 T + 5584059449 T^{2} )^{2} \)
$97$ \( 1 - 3004635070 T^{2} + 73742412689492826049 T^{4} \)
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