Properties

Label 4225.2.a.bh
Level $4225$
Weight $2$
Character orbit 4225.a
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + ( - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + ( - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + (\beta_{2} + 2) q^{11} + ( - \beta_1 + 1) q^{12} + (\beta_{2} - \beta_1 - 1) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + (2 \beta_{2} - 2) q^{17} + 5 \beta_1 q^{18} + \beta_{2} q^{19} + (2 \beta_{2} + 2 \beta_1) q^{21} + (2 \beta_{2} - \beta_1 + 5) q^{22} + (\beta_1 - 5) q^{23} + (3 \beta_{2} + 2 \beta_1 + 2) q^{24} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{27} + (\beta_{2} - \beta_1 + 1) q^{28} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{29} + ( - \beta_{2} - 2 \beta_1 + 4) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} + ( - \beta_{2} - 3 \beta_1 - 1) q^{33} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{34} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{36} + ( - \beta_{2} - 3 \beta_1 + 1) q^{37} + ( - \beta_1 + 3) q^{38} + (2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_1 + 4) q^{42} + (2 \beta_{2} - 3 \beta_1 - 1) q^{43} + (3 \beta_{2} - 4 \beta_1 + 8) q^{44} + ( - 5 \beta_{2} + 2 \beta_1 - 6) q^{46} + (\beta_{2} + \beta_1 + 3) q^{47} + (2 \beta_{2} + 3 \beta_1 + 7) q^{48} + ( - 2 \beta_{2} - 3) q^{49} + ( - 2 \beta_{2} + 4) q^{51} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{53} + ( - 2 \beta_{2} - 4 \beta_1 - 10) q^{54} + ( - \beta_{2} - \beta_1 + 7) q^{56} + ( - \beta_{2} - \beta_1 + 1) q^{57} + (3 \beta_{2} - 3 \beta_1 - 3) q^{58} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{59} + (\beta_{2} + 3 \beta_1 + 1) q^{61} + (4 \beta_{2} - 3 \beta_1 + 3) q^{62} + ( - \beta_{2} - 3 \beta_1 - 5) q^{63} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + ( - \beta_{2} - 5 \beta_1 - 1) q^{66} + (5 \beta_{2} + \beta_1 + 3) q^{67} + ( - 2 \beta_1 + 4) q^{68} + ( - \beta_{2} + 3 \beta_1 + 3) q^{69} + ( - \beta_{2} + 6 \beta_1 + 2) q^{71} + ( - 5 \beta_{2} - 15) q^{72} + (\beta_{2} + 3 \beta_1 - 9) q^{73} + (\beta_{2} - 5 \beta_1 + 1) q^{74} + (\beta_{2} - 2 \beta_1 + 4) q^{76} - 2 \beta_1 q^{77} + (4 \beta_{2} + 2 \beta_1 - 6) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + (2 \beta_{2} - 6 \beta_1 + 10) q^{82} + ( - \beta_{2} + \beta_1 + 7) q^{83} + 2 q^{84} + ( - \beta_{2} - 8 \beta_1 + 8) q^{86} + (6 \beta_{2} + 6 \beta_1) q^{87} + (4 \beta_{2} - 9 \beta_1 + 11) q^{88} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{89} + ( - 6 \beta_{2} + 7 \beta_1 - 13) q^{92} + (3 \beta_{2} + \beta_1 - 1) q^{93} + (3 \beta_{2} + \beta_1 + 5) q^{94} + (\beta_{2} + 6) q^{96} + (6 \beta_{2} + 4 \beta_1 - 6) q^{97} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{98} + (\beta_{2} + 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 4 q^{3} + 5 q^{4} - 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 4 q^{3} + 5 q^{4} - 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} + 2 q^{12} - 4 q^{14} + 5 q^{16} - 6 q^{17} + 5 q^{18} + 2 q^{21} + 14 q^{22} - 14 q^{23} + 8 q^{24} - 10 q^{27} + 2 q^{28} + 6 q^{29} + 10 q^{31} + 11 q^{32} - 6 q^{33} + 10 q^{34} - 11 q^{36} + 8 q^{38} + 4 q^{41} + 14 q^{42} - 6 q^{43} + 20 q^{44} - 16 q^{46} + 10 q^{47} + 24 q^{48} - 9 q^{49} + 12 q^{51} - 8 q^{53} - 34 q^{54} + 20 q^{56} + 2 q^{57} - 12 q^{58} + 8 q^{59} + 6 q^{61} + 6 q^{62} - 18 q^{63} + 33 q^{64} - 8 q^{66} + 10 q^{67} + 10 q^{68} + 12 q^{69} + 12 q^{71} - 45 q^{72} - 24 q^{73} - 2 q^{74} + 10 q^{76} - 2 q^{77} - 16 q^{79} + 23 q^{81} + 24 q^{82} + 22 q^{83} + 6 q^{84} + 16 q^{86} + 6 q^{87} + 24 q^{88} - 10 q^{89} - 32 q^{92} - 2 q^{93} + 16 q^{94} + 18 q^{96} - 14 q^{97} - 25 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
0.311108
2.17009
−1.48119
−1.21432 −1.31111 −0.525428 0 1.59210 2.90321 3.06668 −1.28100 0
1.2 1.53919 −3.17009 0.369102 0 −4.87936 −1.70928 −2.51026 7.04945 0
1.3 2.67513 0.481194 5.15633 0 1.28726 0.806063 8.44358 −2.76845 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(1\)

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 4225.2.a.bh 3
5.b even 2 1 4225.2.a.ba 3
5.c odd 4 2 845.2.b.c 6
13.b even 2 1 325.2.a.j 3
39.d odd 2 1 2925.2.a.bj 3
52.b odd 2 1 5200.2.a.cj 3
65.d even 2 1 325.2.a.k 3
65.f even 4 2 845.2.d.a 6
65.h odd 4 2 65.2.b.a 6
65.k even 4 2 845.2.d.b 6
65.o even 12 4 845.2.l.d 12
65.q odd 12 4 845.2.n.g 12
65.r odd 12 4 845.2.n.f 12
65.t even 12 4 845.2.l.e 12
195.e odd 2 1 2925.2.a.bf 3
195.s even 4 2 585.2.c.b 6
260.g odd 2 1 5200.2.a.cb 3
260.p even 4 2 1040.2.d.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 65.h odd 4 2
325.2.a.j 3 13.b even 2 1
325.2.a.k 3 65.d even 2 1
585.2.c.b 6 195.s even 4 2
845.2.b.c 6 5.c odd 4 2
845.2.d.a 6 65.f even 4 2
845.2.d.b 6 65.k even 4 2
845.2.l.d 12 65.o even 12 4
845.2.l.e 12 65.t even 12 4
845.2.n.f 12 65.r odd 12 4
845.2.n.g 12 65.q odd 12 4
1040.2.d.c 6 260.p even 4 2
2925.2.a.bf 3 195.e odd 2 1
2925.2.a.bj 3 39.d odd 2 1
4225.2.a.ba 3 5.b even 2 1
4225.2.a.bh 3 1.a even 1 1 trivial
5200.2.a.cb 3 260.g odd 2 1
5200.2.a.cj 3 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):

\( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \) Copy content Toggle raw display
\( T_{3}^{3} + 4T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 4T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 6T_{11}^{2} + 8T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T^{2} - T + 5 \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} + 2 T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} + 8 T + 2 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 6 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 4T + 2 \) Copy content Toggle raw display
$23$ \( T^{3} + 14 T^{2} + 62 T + 86 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 36 T + 108 \) Copy content Toggle raw display
$31$ \( T^{3} - 10 T^{2} + 20 T + 26 \) Copy content Toggle raw display
$37$ \( T^{3} - 28T + 52 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 32 T - 32 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 46 T - 278 \) Copy content Toggle raw display
$47$ \( T^{3} - 10 T^{2} + 28 T - 20 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 40 T - 304 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 40 T + 262 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$67$ \( T^{3} - 10 T^{2} - 60 T + 604 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 88 T + 754 \) Copy content Toggle raw display
$73$ \( T^{3} + 24 T^{2} + 164 T + 236 \) Copy content Toggle raw display
$79$ \( T^{3} + 16 T^{2} + 24 T - 16 \) Copy content Toggle raw display
$83$ \( T^{3} - 22 T^{2} + 152 T - 316 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 52 T - 200 \) Copy content Toggle raw display
$97$ \( T^{3} + 14 T^{2} - 84 T - 200 \) Copy content Toggle raw display
show more
show less