Properties

Label 3025.2.a.n
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} + ( 1 + \beta ) q^{4} + 3 q^{6} + ( 2 + \beta ) q^{7} + 3 q^{8} + ( 1 - \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{3} + ( 1 + \beta ) q^{4} + 3 q^{6} + ( 2 + \beta ) q^{7} + 3 q^{8} + ( 1 - \beta ) q^{9} + ( 2 + \beta ) q^{12} + 5 q^{13} + ( 3 + 3 \beta ) q^{14} + ( -2 + \beta ) q^{16} + ( 3 - 3 \beta ) q^{17} -3 q^{18} + q^{19} + ( 1 + 2 \beta ) q^{21} + ( 6 - \beta ) q^{23} + ( -3 + 3 \beta ) q^{24} + 5 \beta q^{26} + ( -1 - 2 \beta ) q^{27} + ( 5 + 4 \beta ) q^{28} + ( 3 + 3 \beta ) q^{29} + ( 5 - 4 \beta ) q^{31} + ( -3 - \beta ) q^{32} -9 q^{34} + ( -2 - \beta ) q^{36} + ( -5 - 2 \beta ) q^{37} + \beta q^{38} + ( -5 + 5 \beta ) q^{39} + ( 3 - 2 \beta ) q^{41} + ( 6 + 3 \beta ) q^{42} + ( 2 - 4 \beta ) q^{43} + ( -3 + 5 \beta ) q^{46} + 3 q^{47} + ( 5 - 2 \beta ) q^{48} + 5 \beta q^{49} + ( -12 + 3 \beta ) q^{51} + ( 5 + 5 \beta ) q^{52} -\beta q^{53} + ( -6 - 3 \beta ) q^{54} + ( 6 + 3 \beta ) q^{56} + ( -1 + \beta ) q^{57} + ( 9 + 6 \beta ) q^{58} + ( -9 + 4 \beta ) q^{59} + ( 4 - 3 \beta ) q^{61} + ( -12 + \beta ) q^{62} + ( -1 - 2 \beta ) q^{63} + ( 1 - 6 \beta ) q^{64} + 4 q^{67} + ( -6 - 3 \beta ) q^{68} + ( -9 + 6 \beta ) q^{69} + 2 \beta q^{71} + ( 3 - 3 \beta ) q^{72} + ( -4 + 3 \beta ) q^{73} + ( -6 - 7 \beta ) q^{74} + ( 1 + \beta ) q^{76} + 15 q^{78} + ( 7 - 3 \beta ) q^{79} + ( -8 + 2 \beta ) q^{81} + ( -6 + \beta ) q^{82} + ( 3 + 5 \beta ) q^{83} + ( 7 + 5 \beta ) q^{84} + ( -12 - 2 \beta ) q^{86} + ( 6 + 3 \beta ) q^{87} + ( 3 + \beta ) q^{89} + ( 10 + 5 \beta ) q^{91} + ( 3 + 4 \beta ) q^{92} + ( -17 + 5 \beta ) q^{93} + 3 \beta q^{94} -3 \beta q^{96} + ( -14 + \beta ) q^{97} + ( 15 + 5 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} + 3 q^{4} + 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9} + O(q^{10}) \) \( 2 q + q^{2} - q^{3} + 3 q^{4} + 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9} + 5 q^{12} + 10 q^{13} + 9 q^{14} - 3 q^{16} + 3 q^{17} - 6 q^{18} + 2 q^{19} + 4 q^{21} + 11 q^{23} - 3 q^{24} + 5 q^{26} - 4 q^{27} + 14 q^{28} + 9 q^{29} + 6 q^{31} - 7 q^{32} - 18 q^{34} - 5 q^{36} - 12 q^{37} + q^{38} - 5 q^{39} + 4 q^{41} + 15 q^{42} - q^{46} + 6 q^{47} + 8 q^{48} + 5 q^{49} - 21 q^{51} + 15 q^{52} - q^{53} - 15 q^{54} + 15 q^{56} - q^{57} + 24 q^{58} - 14 q^{59} + 5 q^{61} - 23 q^{62} - 4 q^{63} - 4 q^{64} + 8 q^{67} - 15 q^{68} - 12 q^{69} + 2 q^{71} + 3 q^{72} - 5 q^{73} - 19 q^{74} + 3 q^{76} + 30 q^{78} + 11 q^{79} - 14 q^{81} - 11 q^{82} + 11 q^{83} + 19 q^{84} - 26 q^{86} + 15 q^{87} + 7 q^{89} + 25 q^{91} + 10 q^{92} - 29 q^{93} + 3 q^{94} - 3 q^{96} - 27 q^{97} + 35 q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
−1.30278 −2.30278 −0.302776 0 3.00000 0.697224 3.00000 2.30278 0
1.2 2.30278 1.30278 3.30278 0 3.00000 4.30278 3.00000 −1.30278 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)

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 3025.2.a.n 2
5.b even 2 1 3025.2.a.h 2
11.b odd 2 1 275.2.a.e 2
33.d even 2 1 2475.2.a.t 2
44.c even 2 1 4400.2.a.bs 2
55.d odd 2 1 275.2.a.f yes 2
55.e even 4 2 275.2.b.c 4
165.d even 2 1 2475.2.a.o 2
165.l odd 4 2 2475.2.c.k 4
220.g even 2 1 4400.2.a.bh 2
220.i odd 4 2 4400.2.b.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 11.b odd 2 1
275.2.a.f yes 2 55.d odd 2 1
275.2.b.c 4 55.e even 4 2
2475.2.a.o 2 165.d even 2 1
2475.2.a.t 2 33.d even 2 1
2475.2.c.k 4 165.l odd 4 2
3025.2.a.h 2 5.b even 2 1
3025.2.a.n 2 1.a even 1 1 trivial
4400.2.a.bh 2 220.g even 2 1
4400.2.a.bs 2 44.c even 2 1
4400.2.b.y 4 220.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{2} - T_{2} - 3 \)
\( T_{3}^{2} + T_{3} - 3 \)
\( T_{19} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 - T + T^{2} \)
$3$ \( -3 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 - 5 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -5 + T )^{2} \)
$17$ \( -27 - 3 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 27 - 11 T + T^{2} \)
$29$ \( -9 - 9 T + T^{2} \)
$31$ \( -43 - 6 T + T^{2} \)
$37$ \( 23 + 12 T + T^{2} \)
$41$ \( -9 - 4 T + T^{2} \)
$43$ \( -52 + T^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( -3 + T + T^{2} \)
$59$ \( -3 + 14 T + T^{2} \)
$61$ \( -23 - 5 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -12 - 2 T + T^{2} \)
$73$ \( -23 + 5 T + T^{2} \)
$79$ \( 1 - 11 T + T^{2} \)
$83$ \( -51 - 11 T + T^{2} \)
$89$ \( 9 - 7 T + T^{2} \)
$97$ \( 179 + 27 T + T^{2} \)
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