Properties

Label 225.4.a.i
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + ( - 6 \beta + 6) q^{7} + ( - \beta - 25) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + ( - 6 \beta + 6) q^{7} + ( - \beta - 25) q^{8} + (6 \beta + 18) q^{11} + ( - 6 \beta + 42) q^{13} + (6 \beta + 54) q^{14} + (3 \beta + 11) q^{16} + ( - 10 \beta - 46) q^{17} + ( - 24 \beta + 40) q^{19} + ( - 30 \beta - 78) q^{22} + ( - 8 \beta + 28) q^{23} + ( - 30 \beta + 18) q^{26} + ( - 18 \beta - 162) q^{28} + ( - 42 \beta + 180) q^{29} + ( - 12 \beta + 32) q^{31} + ( - 9 \beta + 159) q^{32} + (66 \beta + 146) q^{34} + (54 \beta + 126) q^{37} + (8 \beta + 200) q^{38} + (12 \beta + 198) q^{41} + (96 \beta + 12) q^{43} + (90 \beta + 234) q^{44} + ( - 12 \beta + 52) q^{46} + (92 \beta - 136) q^{47} + ( - 36 \beta + 53) q^{49} + (90 \beta - 54) q^{52} + (82 \beta - 242) q^{53} + (150 \beta - 90) q^{56} + ( - 96 \beta + 240) q^{58} + (6 \beta + 90) q^{59} + (96 \beta + 122) q^{61} + ( - 8 \beta + 88) q^{62} + ( - 165 \beta - 157) q^{64} + (60 \beta + 336) q^{67} + ( - 198 \beta - 438) q^{68} + ( - 180 \beta + 108) q^{71} + (108 \beta + 612) q^{73} + ( - 234 \beta - 666) q^{74} + ( - 24 \beta - 600) q^{76} + ( - 108 \beta - 252) q^{77} + (300 \beta + 40) q^{79} + ( - 222 \beta - 318) q^{82} + (208 \beta + 388) q^{83} + ( - 204 \beta - 972) q^{86} + ( - 174 \beta - 510) q^{88} + (144 \beta - 630) q^{89} + ( - 252 \beta + 612) q^{91} + (36 \beta - 156) q^{92} + ( - 48 \beta - 784) q^{94} + ( - 240 \beta - 264) q^{97} + (19 \beta + 307) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8} + 42 q^{11} + 78 q^{13} + 114 q^{14} + 25 q^{16} - 102 q^{17} + 56 q^{19} - 186 q^{22} + 48 q^{23} + 6 q^{26} - 342 q^{28} + 318 q^{29} + 52 q^{31} + 309 q^{32} + 358 q^{34} + 306 q^{37} + 408 q^{38} + 408 q^{41} + 120 q^{43} + 558 q^{44} + 92 q^{46} - 180 q^{47} + 70 q^{49} - 18 q^{52} - 402 q^{53} - 30 q^{56} + 384 q^{58} + 186 q^{59} + 340 q^{61} + 168 q^{62} - 479 q^{64} + 732 q^{67} - 1074 q^{68} + 36 q^{71} + 1332 q^{73} - 1566 q^{74} - 1224 q^{76} - 612 q^{77} + 380 q^{79} - 858 q^{82} + 984 q^{83} - 2148 q^{86} - 1194 q^{88} - 1116 q^{89} + 972 q^{91} - 276 q^{92} - 1616 q^{94} - 768 q^{97} + 633 q^{98}+O(q^{100}) \) 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
3.70156
−2.70156
−4.70156 0 14.1047 0 0 −16.2094 −28.7016 0 0
1.2 1.70156 0 −5.10469 0 0 22.2094 −22.2984 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.i 2
3.b odd 2 1 75.4.a.f 2
5.b even 2 1 225.4.a.o 2
5.c odd 4 2 45.4.b.b 4
12.b even 2 1 1200.4.a.bn 2
15.d odd 2 1 75.4.a.c 2
15.e even 4 2 15.4.b.a 4
20.e even 4 2 720.4.f.j 4
60.h even 2 1 1200.4.a.bt 2
60.l odd 4 2 240.4.f.f 4
120.q odd 4 2 960.4.f.p 4
120.w even 4 2 960.4.f.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 15.e even 4 2
45.4.b.b 4 5.c odd 4 2
75.4.a.c 2 15.d odd 2 1
75.4.a.f 2 3.b odd 2 1
225.4.a.i 2 1.a even 1 1 trivial
225.4.a.o 2 5.b even 2 1
240.4.f.f 4 60.l odd 4 2
720.4.f.j 4 20.e even 4 2
960.4.f.p 4 120.q odd 4 2
960.4.f.q 4 120.w even 4 2
1200.4.a.bn 2 12.b even 2 1
1200.4.a.bt 2 60.h even 2 1

Hecke kernels

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

\( T_{2}^{2} + 3T_{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} - 360 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T - 360 \) Copy content Toggle raw display
$11$ \( T^{2} - 42T + 72 \) Copy content Toggle raw display
$13$ \( T^{2} - 78T + 1152 \) Copy content Toggle raw display
$17$ \( T^{2} + 102T + 1576 \) Copy content Toggle raw display
$19$ \( T^{2} - 56T - 5120 \) Copy content Toggle raw display
$23$ \( T^{2} - 48T - 80 \) Copy content Toggle raw display
$29$ \( T^{2} - 318T + 7200 \) Copy content Toggle raw display
$31$ \( T^{2} - 52T - 800 \) Copy content Toggle raw display
$37$ \( T^{2} - 306T - 6480 \) Copy content Toggle raw display
$41$ \( T^{2} - 408T + 40140 \) Copy content Toggle raw display
$43$ \( T^{2} - 120T - 90864 \) Copy content Toggle raw display
$47$ \( T^{2} + 180T - 78656 \) Copy content Toggle raw display
$53$ \( T^{2} + 402T - 28520 \) Copy content Toggle raw display
$59$ \( T^{2} - 186T + 8280 \) Copy content Toggle raw display
$61$ \( T^{2} - 340T - 65564 \) Copy content Toggle raw display
$67$ \( T^{2} - 732T + 97056 \) Copy content Toggle raw display
$71$ \( T^{2} - 36T - 331776 \) Copy content Toggle raw display
$73$ \( T^{2} - 1332 T + 324000 \) Copy content Toggle raw display
$79$ \( T^{2} - 380T - 886400 \) Copy content Toggle raw display
$83$ \( T^{2} - 984T - 201392 \) Copy content Toggle raw display
$89$ \( T^{2} + 1116T + 98820 \) Copy content Toggle raw display
$97$ \( T^{2} + 768T - 442944 \) Copy content Toggle raw display
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