Properties

Label 225.4.b.d
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} - q^{4} -20 i q^{7} + 21 i q^{8} +O(q^{10})\) \( q + 3 i q^{2} - q^{4} -20 i q^{7} + 21 i q^{8} + 24 q^{11} + 74 i q^{13} + 60 q^{14} -71 q^{16} + 54 i q^{17} + 124 q^{19} + 72 i q^{22} + 120 i q^{23} -222 q^{26} + 20 i q^{28} -78 q^{29} + 200 q^{31} -45 i q^{32} -162 q^{34} + 70 i q^{37} + 372 i q^{38} -330 q^{41} + 92 i q^{43} -24 q^{44} -360 q^{46} -24 i q^{47} -57 q^{49} -74 i q^{52} -450 i q^{53} + 420 q^{56} -234 i q^{58} + 24 q^{59} -322 q^{61} + 600 i q^{62} -433 q^{64} + 196 i q^{67} -54 i q^{68} + 288 q^{71} -430 i q^{73} -210 q^{74} -124 q^{76} -480 i q^{77} + 520 q^{79} -990 i q^{82} -156 i q^{83} -276 q^{86} + 504 i q^{88} + 1026 q^{89} + 1480 q^{91} -120 i q^{92} + 72 q^{94} + 286 i q^{97} -171 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + O(q^{10}) \) \( 2 q - 2 q^{4} + 48 q^{11} + 120 q^{14} - 142 q^{16} + 248 q^{19} - 444 q^{26} - 156 q^{29} + 400 q^{31} - 324 q^{34} - 660 q^{41} - 48 q^{44} - 720 q^{46} - 114 q^{49} + 840 q^{56} + 48 q^{59} - 644 q^{61} - 866 q^{64} + 576 q^{71} - 420 q^{74} - 248 q^{76} + 1040 q^{79} - 552 q^{86} + 2052 q^{89} + 2960 q^{91} + 144 q^{94} + O(q^{100}) \)

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
3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 0 0
199.2 3.00000i 0 −1.00000 0 0 20.0000i 21.0000i 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.d 2
3.b odd 2 1 75.4.b.a 2
5.b even 2 1 inner 225.4.b.d 2
5.c odd 4 1 45.4.a.b 1
5.c odd 4 1 225.4.a.g 1
12.b even 2 1 1200.4.f.m 2
15.d odd 2 1 75.4.b.a 2
15.e even 4 1 15.4.a.b 1
15.e even 4 1 75.4.a.a 1
20.e even 4 1 720.4.a.r 1
35.f even 4 1 2205.4.a.c 1
45.k odd 12 2 405.4.e.k 2
45.l even 12 2 405.4.e.d 2
60.h even 2 1 1200.4.f.m 2
60.l odd 4 1 240.4.a.f 1
60.l odd 4 1 1200.4.a.o 1
105.k odd 4 1 735.4.a.i 1
120.q odd 4 1 960.4.a.l 1
120.w even 4 1 960.4.a.bi 1
165.l odd 4 1 1815.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 15.e even 4 1
45.4.a.b 1 5.c odd 4 1
75.4.a.a 1 15.e even 4 1
75.4.b.a 2 3.b odd 2 1
75.4.b.a 2 15.d odd 2 1
225.4.a.g 1 5.c odd 4 1
225.4.b.d 2 1.a even 1 1 trivial
225.4.b.d 2 5.b even 2 1 inner
240.4.a.f 1 60.l odd 4 1
405.4.e.d 2 45.l even 12 2
405.4.e.k 2 45.k odd 12 2
720.4.a.r 1 20.e even 4 1
735.4.a.i 1 105.k odd 4 1
960.4.a.l 1 120.q odd 4 1
960.4.a.bi 1 120.w even 4 1
1200.4.a.o 1 60.l odd 4 1
1200.4.f.m 2 12.b even 2 1
1200.4.f.m 2 60.h even 2 1
1815.4.a.a 1 165.l odd 4 1
2205.4.a.c 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 \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 9 \)
\( T_{7}^{2} + 400 \)
\( T_{11} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 400 + T^{2} \)
$11$ \( ( -24 + T )^{2} \)
$13$ \( 5476 + T^{2} \)
$17$ \( 2916 + T^{2} \)
$19$ \( ( -124 + T )^{2} \)
$23$ \( 14400 + T^{2} \)
$29$ \( ( 78 + T )^{2} \)
$31$ \( ( -200 + T )^{2} \)
$37$ \( 4900 + T^{2} \)
$41$ \( ( 330 + T )^{2} \)
$43$ \( 8464 + T^{2} \)
$47$ \( 576 + T^{2} \)
$53$ \( 202500 + T^{2} \)
$59$ \( ( -24 + T )^{2} \)
$61$ \( ( 322 + T )^{2} \)
$67$ \( 38416 + T^{2} \)
$71$ \( ( -288 + T )^{2} \)
$73$ \( 184900 + T^{2} \)
$79$ \( ( -520 + T )^{2} \)
$83$ \( 24336 + T^{2} \)
$89$ \( ( -1026 + T )^{2} \)
$97$ \( 81796 + T^{2} \)
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