Properties

Label 225.4.b.f
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 25)
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.

\(f(q)\) \(=\) \( q + i q^{2} + 7 q^{4} -6 i q^{7} + 15 i q^{8} +O(q^{10})\) \( q + i q^{2} + 7 q^{4} -6 i q^{7} + 15 i q^{8} + 43 q^{11} -28 i q^{13} + 6 q^{14} + 41 q^{16} + 91 i q^{17} + 35 q^{19} + 43 i q^{22} -162 i q^{23} + 28 q^{26} -42 i q^{28} + 160 q^{29} + 42 q^{31} + 161 i q^{32} -91 q^{34} + 314 i q^{37} + 35 i q^{38} + 203 q^{41} + 92 i q^{43} + 301 q^{44} + 162 q^{46} + 196 i q^{47} + 307 q^{49} -196 i q^{52} -82 i q^{53} + 90 q^{56} + 160 i q^{58} -280 q^{59} -518 q^{61} + 42 i q^{62} + 167 q^{64} -141 i q^{67} + 637 i q^{68} -412 q^{71} -763 i q^{73} -314 q^{74} + 245 q^{76} -258 i q^{77} -510 q^{79} + 203 i q^{82} -777 i q^{83} -92 q^{86} + 645 i q^{88} -945 q^{89} -168 q^{91} -1134 i q^{92} -196 q^{94} -1246 i q^{97} + 307 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + O(q^{10}) \) \( 2 q + 14 q^{4} + 86 q^{11} + 12 q^{14} + 82 q^{16} + 70 q^{19} + 56 q^{26} + 320 q^{29} + 84 q^{31} - 182 q^{34} + 406 q^{41} + 602 q^{44} + 324 q^{46} + 614 q^{49} + 180 q^{56} - 560 q^{59} - 1036 q^{61} + 334 q^{64} - 824 q^{71} - 628 q^{74} + 490 q^{76} - 1020 q^{79} - 184 q^{86} - 1890 q^{89} - 336 q^{91} - 392 q^{94} + O(q^{100}) \)

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
1.00000i 0 7.00000 0 0 6.00000i 15.0000i 0 0
199.2 1.00000i 0 7.00000 0 0 6.00000i 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.f 2
3.b odd 2 1 25.4.b.b 2
5.b even 2 1 inner 225.4.b.f 2
5.c odd 4 1 225.4.a.c 1
5.c odd 4 1 225.4.a.e 1
12.b even 2 1 400.4.c.e 2
15.d odd 2 1 25.4.b.b 2
15.e even 4 1 25.4.a.a 1
15.e even 4 1 25.4.a.b yes 1
60.h even 2 1 400.4.c.e 2
60.l odd 4 1 400.4.a.c 1
60.l odd 4 1 400.4.a.s 1
105.k odd 4 1 1225.4.a.h 1
105.k odd 4 1 1225.4.a.i 1
120.q odd 4 1 1600.4.a.h 1
120.q odd 4 1 1600.4.a.bs 1
120.w even 4 1 1600.4.a.i 1
120.w even 4 1 1600.4.a.bt 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 15.e even 4 1
25.4.a.b yes 1 15.e even 4 1
25.4.b.b 2 3.b odd 2 1
25.4.b.b 2 15.d odd 2 1
225.4.a.c 1 5.c odd 4 1
225.4.a.e 1 5.c odd 4 1
225.4.b.f 2 1.a even 1 1 trivial
225.4.b.f 2 5.b even 2 1 inner
400.4.a.c 1 60.l odd 4 1
400.4.a.s 1 60.l odd 4 1
400.4.c.e 2 12.b even 2 1
400.4.c.e 2 60.h even 2 1
1225.4.a.h 1 105.k odd 4 1
1225.4.a.i 1 105.k odd 4 1
1600.4.a.h 1 120.q odd 4 1
1600.4.a.i 1 120.w even 4 1
1600.4.a.bs 1 120.q odd 4 1
1600.4.a.bt 1 120.w even 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 36 \)
\( T_{11} - 43 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 36 + T^{2} \)
$11$ \( ( -43 + T )^{2} \)
$13$ \( 784 + T^{2} \)
$17$ \( 8281 + T^{2} \)
$19$ \( ( -35 + T )^{2} \)
$23$ \( 26244 + T^{2} \)
$29$ \( ( -160 + T )^{2} \)
$31$ \( ( -42 + T )^{2} \)
$37$ \( 98596 + T^{2} \)
$41$ \( ( -203 + T )^{2} \)
$43$ \( 8464 + T^{2} \)
$47$ \( 38416 + T^{2} \)
$53$ \( 6724 + T^{2} \)
$59$ \( ( 280 + T )^{2} \)
$61$ \( ( 518 + T )^{2} \)
$67$ \( 19881 + T^{2} \)
$71$ \( ( 412 + T )^{2} \)
$73$ \( 582169 + T^{2} \)
$79$ \( ( 510 + T )^{2} \)
$83$ \( 603729 + T^{2} \)
$89$ \( ( 945 + T )^{2} \)
$97$ \( 1552516 + T^{2} \)
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