Properties

Label 225.4.b.c
Level $225$
Weight $4$
Character orbit 225.b
Analytic conductor $13.275$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} - 8 q^{4} + 3 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{2} - 8 q^{4} + 3 \beta q^{7} - 32 q^{11} + 19 \beta q^{13} - 24 q^{14} - 64 q^{16} - 13 \beta q^{17} - 100 q^{19} - 64 \beta q^{22} - 39 \beta q^{23} - 152 q^{26} - 24 \beta q^{28} - 50 q^{29} - 108 q^{31} - 128 \beta q^{32} + 104 q^{34} + 133 \beta q^{37} - 200 \beta q^{38} - 22 q^{41} - 221 \beta q^{43} + 256 q^{44} + 312 q^{46} + 257 \beta q^{47} + 307 q^{49} - 152 \beta q^{52} + \beta q^{53} - 100 \beta q^{58} + 500 q^{59} - 518 q^{61} - 216 \beta q^{62} + 512 q^{64} + 63 \beta q^{67} + 104 \beta q^{68} - 412 q^{71} + 439 \beta q^{73} - 1064 q^{74} + 800 q^{76} - 96 \beta q^{77} - 600 q^{79} - 44 \beta q^{82} + 141 \beta q^{83} + 1768 q^{86} - 150 q^{89} - 228 q^{91} + 312 \beta q^{92} - 2056 q^{94} + 193 \beta q^{97} + 614 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{4} - 64 q^{11} - 48 q^{14} - 128 q^{16} - 200 q^{19} - 304 q^{26} - 100 q^{29} - 216 q^{31} + 208 q^{34} - 44 q^{41} + 512 q^{44} + 624 q^{46} + 614 q^{49} + 1000 q^{59} - 1036 q^{61} + 1024 q^{64} - 824 q^{71} - 2128 q^{74} + 1600 q^{76} - 1200 q^{79} + 3536 q^{86} - 300 q^{89} - 456 q^{91} - 4112 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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
4.00000i 0 −8.00000 0 0 6.00000i 0 0 0
199.2 4.00000i 0 −8.00000 0 0 6.00000i 0 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 225.4.b.c 2
3.b odd 2 1 25.4.b.a 2
5.b even 2 1 inner 225.4.b.c 2
5.c odd 4 1 45.4.a.d 1
5.c odd 4 1 225.4.a.b 1
12.b even 2 1 400.4.c.k 2
15.d odd 2 1 25.4.b.a 2
15.e even 4 1 5.4.a.a 1
15.e even 4 1 25.4.a.c 1
20.e even 4 1 720.4.a.u 1
35.f even 4 1 2205.4.a.q 1
45.k odd 12 2 405.4.e.c 2
45.l even 12 2 405.4.e.l 2
60.h even 2 1 400.4.c.k 2
60.l odd 4 1 80.4.a.d 1
60.l odd 4 1 400.4.a.m 1
105.k odd 4 1 245.4.a.a 1
105.k odd 4 1 1225.4.a.k 1
105.w odd 12 2 245.4.e.g 2
105.x even 12 2 245.4.e.f 2
120.q odd 4 1 320.4.a.h 1
120.q odd 4 1 1600.4.a.s 1
120.w even 4 1 320.4.a.g 1
120.w even 4 1 1600.4.a.bi 1
165.l odd 4 1 605.4.a.d 1
195.s even 4 1 845.4.a.b 1
240.z odd 4 1 1280.4.d.l 2
240.bb even 4 1 1280.4.d.e 2
240.bd odd 4 1 1280.4.d.l 2
240.bf even 4 1 1280.4.d.e 2
255.o even 4 1 1445.4.a.a 1
285.j odd 4 1 1805.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 15.e even 4 1
25.4.a.c 1 15.e even 4 1
25.4.b.a 2 3.b odd 2 1
25.4.b.a 2 15.d odd 2 1
45.4.a.d 1 5.c odd 4 1
80.4.a.d 1 60.l odd 4 1
225.4.a.b 1 5.c odd 4 1
225.4.b.c 2 1.a even 1 1 trivial
225.4.b.c 2 5.b even 2 1 inner
245.4.a.a 1 105.k odd 4 1
245.4.e.f 2 105.x even 12 2
245.4.e.g 2 105.w odd 12 2
320.4.a.g 1 120.w even 4 1
320.4.a.h 1 120.q odd 4 1
400.4.a.m 1 60.l odd 4 1
400.4.c.k 2 12.b even 2 1
400.4.c.k 2 60.h even 2 1
405.4.e.c 2 45.k odd 12 2
405.4.e.l 2 45.l even 12 2
605.4.a.d 1 165.l odd 4 1
720.4.a.u 1 20.e even 4 1
845.4.a.b 1 195.s even 4 1
1225.4.a.k 1 105.k odd 4 1
1280.4.d.e 2 240.bb even 4 1
1280.4.d.e 2 240.bf even 4 1
1280.4.d.l 2 240.z odd 4 1
1280.4.d.l 2 240.bd odd 4 1
1445.4.a.a 1 255.o even 4 1
1600.4.a.s 1 120.q odd 4 1
1600.4.a.bi 1 120.w even 4 1
1805.4.a.h 1 285.j odd 4 1
2205.4.a.q 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{11} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1444 \) Copy content Toggle raw display
$17$ \( T^{2} + 676 \) Copy content Toggle raw display
$19$ \( (T + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6084 \) Copy content Toggle raw display
$29$ \( (T + 50)^{2} \) Copy content Toggle raw display
$31$ \( (T + 108)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 70756 \) Copy content Toggle raw display
$41$ \( (T + 22)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 195364 \) Copy content Toggle raw display
$47$ \( T^{2} + 264196 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T - 500)^{2} \) Copy content Toggle raw display
$61$ \( (T + 518)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 15876 \) Copy content Toggle raw display
$71$ \( (T + 412)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 770884 \) Copy content Toggle raw display
$79$ \( (T + 600)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 79524 \) Copy content Toggle raw display
$89$ \( (T + 150)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 148996 \) Copy content Toggle raw display
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