Properties

Label 225.4.b.e
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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + i q^{2} + 7 q^{4} + 24 i q^{7} + 15 i q^{8} +O(q^{10})\) \( q + i q^{2} + 7 q^{4} + 24 i q^{7} + 15 i q^{8} -52 q^{11} + 22 i q^{13} -24 q^{14} + 41 q^{16} -14 i q^{17} + 20 q^{19} -52 i q^{22} + 168 i q^{23} -22 q^{26} + 168 i q^{28} + 230 q^{29} -288 q^{31} + 161 i q^{32} + 14 q^{34} + 34 i q^{37} + 20 i q^{38} -122 q^{41} -188 i q^{43} -364 q^{44} -168 q^{46} + 256 i q^{47} -233 q^{49} + 154 i q^{52} + 338 i q^{53} -360 q^{56} + 230 i q^{58} + 100 q^{59} + 742 q^{61} -288 i q^{62} + 167 q^{64} + 84 i q^{67} -98 i q^{68} + 328 q^{71} -38 i q^{73} -34 q^{74} + 140 q^{76} -1248 i q^{77} + 240 q^{79} -122 i q^{82} -1212 i q^{83} + 188 q^{86} -780 i q^{88} + 330 q^{89} -528 q^{91} + 1176 i q^{92} -256 q^{94} -866 i q^{97} -233 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{4} + O(q^{10}) \) \( 2q + 14q^{4} - 104q^{11} - 48q^{14} + 82q^{16} + 40q^{19} - 44q^{26} + 460q^{29} - 576q^{31} + 28q^{34} - 244q^{41} - 728q^{44} - 336q^{46} - 466q^{49} - 720q^{56} + 200q^{59} + 1484q^{61} + 334q^{64} + 656q^{71} - 68q^{74} + 280q^{76} + 480q^{79} + 376q^{86} + 660q^{89} - 1056q^{91} - 512q^{94} + O(q^{100}) \)

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
1.00000i 0 7.00000 0 0 24.0000i 15.0000i 0 0
199.2 1.00000i 0 7.00000 0 0 24.0000i 15.0000i 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.e 2
3.b odd 2 1 75.4.b.b 2
5.b even 2 1 inner 225.4.b.e 2
5.c odd 4 1 45.4.a.c 1
5.c odd 4 1 225.4.a.f 1
12.b even 2 1 1200.4.f.b 2
15.d odd 2 1 75.4.b.b 2
15.e even 4 1 15.4.a.a 1
15.e even 4 1 75.4.a.b 1
20.e even 4 1 720.4.a.n 1
35.f even 4 1 2205.4.a.l 1
45.k odd 12 2 405.4.e.i 2
45.l even 12 2 405.4.e.g 2
60.h even 2 1 1200.4.f.b 2
60.l odd 4 1 240.4.a.e 1
60.l odd 4 1 1200.4.a.t 1
105.k odd 4 1 735.4.a.e 1
120.q odd 4 1 960.4.a.ba 1
120.w even 4 1 960.4.a.b 1
165.l odd 4 1 1815.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 15.e even 4 1
45.4.a.c 1 5.c odd 4 1
75.4.a.b 1 15.e even 4 1
75.4.b.b 2 3.b odd 2 1
75.4.b.b 2 15.d odd 2 1
225.4.a.f 1 5.c odd 4 1
225.4.b.e 2 1.a even 1 1 trivial
225.4.b.e 2 5.b even 2 1 inner
240.4.a.e 1 60.l odd 4 1
405.4.e.g 2 45.l even 12 2
405.4.e.i 2 45.k odd 12 2
720.4.a.n 1 20.e even 4 1
735.4.a.e 1 105.k odd 4 1
960.4.a.b 1 120.w even 4 1
960.4.a.ba 1 120.q odd 4 1
1200.4.a.t 1 60.l odd 4 1
1200.4.f.b 2 12.b even 2 1
1200.4.f.b 2 60.h even 2 1
1815.4.a.e 1 165.l odd 4 1
2205.4.a.l 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 \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 576 \)
\( T_{11} + 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 576 + T^{2} \)
$11$ \( ( 52 + T )^{2} \)
$13$ \( 484 + T^{2} \)
$17$ \( 196 + T^{2} \)
$19$ \( ( -20 + T )^{2} \)
$23$ \( 28224 + T^{2} \)
$29$ \( ( -230 + T )^{2} \)
$31$ \( ( 288 + T )^{2} \)
$37$ \( 1156 + T^{2} \)
$41$ \( ( 122 + T )^{2} \)
$43$ \( 35344 + T^{2} \)
$47$ \( 65536 + T^{2} \)
$53$ \( 114244 + T^{2} \)
$59$ \( ( -100 + T )^{2} \)
$61$ \( ( -742 + T )^{2} \)
$67$ \( 7056 + T^{2} \)
$71$ \( ( -328 + T )^{2} \)
$73$ \( 1444 + T^{2} \)
$79$ \( ( -240 + T )^{2} \)
$83$ \( 1468944 + T^{2} \)
$89$ \( ( -330 + T )^{2} \)
$97$ \( 749956 + T^{2} \)
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