Properties

Label 3025.2.a.o
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + 2 \beta q^{3} + ( 1 + 2 \beta ) q^{4} + ( 4 + 2 \beta ) q^{6} -2 q^{7} + ( 3 + \beta ) q^{8} + 5 q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + 2 \beta q^{3} + ( 1 + 2 \beta ) q^{4} + ( 4 + 2 \beta ) q^{6} -2 q^{7} + ( 3 + \beta ) q^{8} + 5 q^{9} + ( 8 + 2 \beta ) q^{12} + ( -4 + 2 \beta ) q^{13} + ( -2 - 2 \beta ) q^{14} + 3 q^{16} + ( 4 + 2 \beta ) q^{17} + ( 5 + 5 \beta ) q^{18} -4 \beta q^{21} + 2 \beta q^{23} + ( 4 + 6 \beta ) q^{24} -2 \beta q^{26} + 4 \beta q^{27} + ( -2 - 4 \beta ) q^{28} + ( -2 + 4 \beta ) q^{29} + ( -3 + \beta ) q^{32} + ( 8 + 6 \beta ) q^{34} + ( 5 + 10 \beta ) q^{36} + ( 2 + 4 \beta ) q^{37} + ( 8 - 8 \beta ) q^{39} -6 q^{41} + ( -8 - 4 \beta ) q^{42} -6 q^{43} + ( 4 + 2 \beta ) q^{46} -2 \beta q^{47} + 6 \beta q^{48} -3 q^{49} + ( 8 + 8 \beta ) q^{51} + ( 4 - 6 \beta ) q^{52} + ( -6 - 4 \beta ) q^{53} + ( 8 + 4 \beta ) q^{54} + ( -6 - 2 \beta ) q^{56} + ( 6 + 2 \beta ) q^{58} + ( -4 + 4 \beta ) q^{59} + ( -2 + 8 \beta ) q^{61} -10 q^{63} + ( -7 - 2 \beta ) q^{64} + ( -4 - 6 \beta ) q^{67} + ( 12 + 10 \beta ) q^{68} + 8 q^{69} + 8 \beta q^{71} + ( 15 + 5 \beta ) q^{72} + ( -4 + 2 \beta ) q^{73} + ( 10 + 6 \beta ) q^{74} -8 q^{78} -4 q^{79} + q^{81} + ( -6 - 6 \beta ) q^{82} -6 q^{83} + ( -16 - 4 \beta ) q^{84} + ( -6 - 6 \beta ) q^{86} + ( 16 - 4 \beta ) q^{87} + ( -2 - 8 \beta ) q^{89} + ( 8 - 4 \beta ) q^{91} + ( 8 + 2 \beta ) q^{92} + ( -4 - 2 \beta ) q^{94} + ( 4 - 6 \beta ) q^{96} + ( 2 - 4 \beta ) q^{97} + ( -3 - 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9} + 16 q^{12} - 8 q^{13} - 4 q^{14} + 6 q^{16} + 8 q^{17} + 10 q^{18} + 8 q^{24} - 4 q^{28} - 4 q^{29} - 6 q^{32} + 16 q^{34} + 10 q^{36} + 4 q^{37} + 16 q^{39} - 12 q^{41} - 16 q^{42} - 12 q^{43} + 8 q^{46} - 6 q^{49} + 16 q^{51} + 8 q^{52} - 12 q^{53} + 16 q^{54} - 12 q^{56} + 12 q^{58} - 8 q^{59} - 4 q^{61} - 20 q^{63} - 14 q^{64} - 8 q^{67} + 24 q^{68} + 16 q^{69} + 30 q^{72} - 8 q^{73} + 20 q^{74} - 16 q^{78} - 8 q^{79} + 2 q^{81} - 12 q^{82} - 12 q^{83} - 32 q^{84} - 12 q^{86} + 32 q^{87} - 4 q^{89} + 16 q^{91} + 16 q^{92} - 8 q^{94} + 8 q^{96} + 4 q^{97} - 6 q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−0.414214 −2.82843 −1.82843 0 1.17157 −2.00000 1.58579 5.00000 0
1.2 2.41421 2.82843 3.82843 0 6.82843 −2.00000 4.41421 5.00000 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.o 2
5.b even 2 1 605.2.a.d 2
11.b odd 2 1 275.2.a.c 2
15.d odd 2 1 5445.2.a.y 2
20.d odd 2 1 9680.2.a.bn 2
33.d even 2 1 2475.2.a.x 2
44.c even 2 1 4400.2.a.bn 2
55.d odd 2 1 55.2.a.b 2
55.e even 4 2 275.2.b.d 4
55.h odd 10 4 605.2.g.f 8
55.j even 10 4 605.2.g.l 8
165.d even 2 1 495.2.a.b 2
165.l odd 4 2 2475.2.c.l 4
220.g even 2 1 880.2.a.m 2
220.i odd 4 2 4400.2.b.q 4
385.h even 2 1 2695.2.a.f 2
440.c even 2 1 3520.2.a.bo 2
440.o odd 2 1 3520.2.a.bn 2
660.g odd 2 1 7920.2.a.ch 2
715.c odd 2 1 9295.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 55.d odd 2 1
275.2.a.c 2 11.b odd 2 1
275.2.b.d 4 55.e even 4 2
495.2.a.b 2 165.d even 2 1
605.2.a.d 2 5.b even 2 1
605.2.g.f 8 55.h odd 10 4
605.2.g.l 8 55.j even 10 4
880.2.a.m 2 220.g even 2 1
2475.2.a.x 2 33.d even 2 1
2475.2.c.l 4 165.l odd 4 2
2695.2.a.f 2 385.h even 2 1
3025.2.a.o 2 1.a even 1 1 trivial
3520.2.a.bn 2 440.o odd 2 1
3520.2.a.bo 2 440.c even 2 1
4400.2.a.bn 2 44.c even 2 1
4400.2.b.q 4 220.i odd 4 2
5445.2.a.y 2 15.d odd 2 1
7920.2.a.ch 2 660.g odd 2 1
9295.2.a.g 2 715.c odd 2 1
9680.2.a.bn 2 20.d odd 2 1

Hecke kernels

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

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{3}^{2} - 8 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 2 T + T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 8 + 8 T + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( -28 + 4 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( -28 - 4 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( 4 + 12 T + T^{2} \)
$59$ \( -16 + 8 T + T^{2} \)
$61$ \( -124 + 4 T + T^{2} \)
$67$ \( -56 + 8 T + T^{2} \)
$71$ \( -128 + T^{2} \)
$73$ \( 8 + 8 T + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( -124 + 4 T + T^{2} \)
$97$ \( -28 - 4 T + T^{2} \)
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