Properties

Label 225.4.a.j
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2754297513\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 12 - 2 \beta ) q^{4} + ( -13 - 4 \beta ) q^{7} + ( -42 + 6 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 12 - 2 \beta ) q^{4} + ( -13 - 4 \beta ) q^{7} + ( -42 + 6 \beta ) q^{8} + ( -14 - 4 \beta ) q^{11} + ( -9 + 16 \beta ) q^{13} + ( -63 - 9 \beta ) q^{14} + ( 60 - 32 \beta ) q^{16} + ( -34 - 20 \beta ) q^{17} + ( 3 - 4 \beta ) q^{19} + ( -62 - 10 \beta ) q^{22} + ( 66 - 12 \beta ) q^{23} + ( 313 - 25 \beta ) q^{26} + ( -4 - 22 \beta ) q^{28} + ( -46 + 28 \beta ) q^{29} + ( 61 + 28 \beta ) q^{31} + ( -332 + 44 \beta ) q^{32} + ( -346 - 14 \beta ) q^{34} + ( 142 - 24 \beta ) q^{37} + ( -79 + 7 \beta ) q^{38} + ( -196 + 52 \beta ) q^{41} + ( -345 + 4 \beta ) q^{43} + ( -16 - 20 \beta ) q^{44} + ( -294 + 78 \beta ) q^{46} + ( -310 - 32 \beta ) q^{47} + ( 130 + 104 \beta ) q^{49} + ( -716 + 210 \beta ) q^{52} + ( -424 + 28 \beta ) q^{53} + ( 90 + 90 \beta ) q^{56} + ( 578 - 74 \beta ) q^{58} + ( -62 - 64 \beta ) q^{59} + ( 375 + 56 \beta ) q^{61} + ( 471 + 33 \beta ) q^{62} + ( 688 - 120 \beta ) q^{64} + ( 179 - 100 \beta ) q^{67} + ( 352 - 172 \beta ) q^{68} + ( -412 - 20 \beta ) q^{71} + ( 54 - 8 \beta ) q^{73} + ( -598 + 166 \beta ) q^{74} + ( 188 - 54 \beta ) q^{76} + ( 486 + 108 \beta ) q^{77} + ( -440 + 160 \beta ) q^{79} + ( 1184 - 248 \beta ) q^{82} + ( 78 + 192 \beta ) q^{83} + ( 421 - 349 \beta ) q^{86} + ( 132 + 84 \beta ) q^{88} + ( 432 + 144 \beta ) q^{89} + ( -1099 - 172 \beta ) q^{91} + ( 1248 - 276 \beta ) q^{92} + ( -298 - 278 \beta ) q^{94} -521 q^{97} + ( 1846 + 26 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8} - 28 q^{11} - 18 q^{13} - 126 q^{14} + 120 q^{16} - 68 q^{17} + 6 q^{19} - 124 q^{22} + 132 q^{23} + 626 q^{26} - 8 q^{28} - 92 q^{29} + 122 q^{31} - 664 q^{32} - 692 q^{34} + 284 q^{37} - 158 q^{38} - 392 q^{41} - 690 q^{43} - 32 q^{44} - 588 q^{46} - 620 q^{47} + 260 q^{49} - 1432 q^{52} - 848 q^{53} + 180 q^{56} + 1156 q^{58} - 124 q^{59} + 750 q^{61} + 942 q^{62} + 1376 q^{64} + 358 q^{67} + 704 q^{68} - 824 q^{71} + 108 q^{73} - 1196 q^{74} + 376 q^{76} + 972 q^{77} - 880 q^{79} + 2368 q^{82} + 156 q^{83} + 842 q^{86} + 264 q^{88} + 864 q^{89} - 2198 q^{91} + 2496 q^{92} - 596 q^{94} - 1042 q^{97} + 3692 q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.35890
4.35890
−5.35890 0 20.7178 0 0 4.43560 −68.1534 0 0
1.2 3.35890 0 3.28220 0 0 −30.4356 −15.8466 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.a.j 2
3.b odd 2 1 75.4.a.e yes 2
5.b even 2 1 225.4.a.n 2
5.c odd 4 2 225.4.b.h 4
12.b even 2 1 1200.4.a.bu 2
15.d odd 2 1 75.4.a.d 2
15.e even 4 2 75.4.b.c 4
60.h even 2 1 1200.4.a.bl 2
60.l odd 4 2 1200.4.f.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 15.d odd 2 1
75.4.a.e yes 2 3.b odd 2 1
75.4.b.c 4 15.e even 4 2
225.4.a.j 2 1.a even 1 1 trivial
225.4.a.n 2 5.b even 2 1
225.4.b.h 4 5.c odd 4 2
1200.4.a.bl 2 60.h even 2 1
1200.4.a.bu 2 12.b even 2 1
1200.4.f.v 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} + 2 T_{2} - 18 \)
\( T_{7}^{2} + 26 T_{7} - 135 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -18 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -135 + 26 T + T^{2} \)
$11$ \( -108 + 28 T + T^{2} \)
$13$ \( -4783 + 18 T + T^{2} \)
$17$ \( -6444 + 68 T + T^{2} \)
$19$ \( -295 - 6 T + T^{2} \)
$23$ \( 1620 - 132 T + T^{2} \)
$29$ \( -12780 + 92 T + T^{2} \)
$31$ \( -11175 - 122 T + T^{2} \)
$37$ \( 9220 - 284 T + T^{2} \)
$41$ \( -12960 + 392 T + T^{2} \)
$43$ \( 118721 + 690 T + T^{2} \)
$47$ \( 76644 + 620 T + T^{2} \)
$53$ \( 164880 + 848 T + T^{2} \)
$59$ \( -73980 + 124 T + T^{2} \)
$61$ \( 81041 - 750 T + T^{2} \)
$67$ \( -157959 - 358 T + T^{2} \)
$71$ \( 162144 + 824 T + T^{2} \)
$73$ \( 1700 - 108 T + T^{2} \)
$79$ \( -292800 + 880 T + T^{2} \)
$83$ \( -694332 - 156 T + T^{2} \)
$89$ \( -207360 - 864 T + T^{2} \)
$97$ \( ( 521 + T )^{2} \)
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