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: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} -8 q^{4} + 6 i q^{7} +O(q^{10})\) \( q + 4 i q^{2} -8 q^{4} + 6 i q^{7} -32 q^{11} + 38 i q^{13} -24 q^{14} -64 q^{16} -26 i q^{17} -100 q^{19} -128 i q^{22} -78 i q^{23} -152 q^{26} -48 i q^{28} -50 q^{29} -108 q^{31} -256 i q^{32} + 104 q^{34} + 266 i q^{37} -400 i q^{38} -22 q^{41} -442 i q^{43} + 256 q^{44} + 312 q^{46} + 514 i q^{47} + 307 q^{49} -304 i q^{52} + 2 i q^{53} -200 i q^{58} + 500 q^{59} -518 q^{61} -432 i q^{62} + 512 q^{64} + 126 i q^{67} + 208 i q^{68} -412 q^{71} + 878 i q^{73} -1064 q^{74} + 800 q^{76} -192 i q^{77} -600 q^{79} -88 i q^{82} + 282 i q^{83} + 1768 q^{86} -150 q^{89} -228 q^{91} + 624 i q^{92} -2056 q^{94} + 386 i q^{97} + 1228 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{4} + O(q^{10}) \) \( 2q - 16q^{4} - 64q^{11} - 48q^{14} - 128q^{16} - 200q^{19} - 304q^{26} - 100q^{29} - 216q^{31} + 208q^{34} - 44q^{41} + 512q^{44} + 624q^{46} + 614q^{49} + 1000q^{59} - 1036q^{61} + 1024q^{64} - 824q^{71} - 2128q^{74} + 1600q^{76} - 1200q^{79} + 3536q^{86} - 300q^{89} - 456q^{91} - 4112q^{94} + O(q^{100}) \)

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
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 \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 16 \)
\( T_{7}^{2} + 36 \)
\( T_{11} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T + 8 T^{2} )( 1 + 4 T + 8 T^{2} ) \)
$3$ 1
$5$ 1
$7$ \( 1 - 650 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 + 32 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 2950 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 9150 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 + 100 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 18250 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 + 50 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 108 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 30550 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 + 22 T + 68921 T^{2} )^{2} \)
$43$ \( 1 + 36350 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 + 56550 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 297750 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 - 500 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 + 518 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 585650 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 412 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 7150 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 + 600 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 1064050 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 + 150 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1676350 T^{2} + 832972004929 T^{4} \)
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