Properties

Label 225.4.a.n
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
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.

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

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
−3.35890 0 3.28220 0 0 30.4356 15.8466 0 0
1.2 5.35890 0 20.7178 0 0 −4.43560 68.1534 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.n 2
3.b odd 2 1 75.4.a.d 2
5.b even 2 1 225.4.a.j 2
5.c odd 4 2 225.4.b.h 4
12.b even 2 1 1200.4.a.bl 2
15.d odd 2 1 75.4.a.e yes 2
15.e even 4 2 75.4.b.c 4
60.h even 2 1 1200.4.a.bu 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 3.b odd 2 1
75.4.a.e yes 2 15.d odd 2 1
75.4.b.c 4 15.e even 4 2
225.4.a.j 2 5.b even 2 1
225.4.a.n 2 1.a even 1 1 trivial
225.4.b.h 4 5.c odd 4 2
1200.4.a.bl 2 12.b even 2 1
1200.4.a.bu 2 60.h 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 2T_{2} - 18 \) Copy content Toggle raw display
\( T_{7}^{2} - 26T_{7} - 135 \) Copy content Toggle raw display

Hecke characteristic polynomials

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