Properties

Label 9295.2.a.g
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9295,2,Mod(1,9295)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9295, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9295.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9295 = 5 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9295.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.2209486788\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + 2 \beta q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - 2 \beta + 4) q^{6} + 2 q^{7} + (\beta - 3) q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + 2 \beta q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - 2 \beta + 4) q^{6} + 2 q^{7} + (\beta - 3) q^{8} + 5 q^{9} + (\beta - 1) q^{10} - q^{11} + (2 \beta - 8) q^{12} + (2 \beta - 2) q^{14} + 2 \beta q^{15} + 3 q^{16} + ( - 2 \beta + 4) q^{17} + (5 \beta - 5) q^{18} + ( - 2 \beta + 1) q^{20} + 4 \beta q^{21} + ( - \beta + 1) q^{22} + 2 \beta q^{23} + ( - 6 \beta + 4) q^{24} + q^{25} + 4 \beta q^{27} + ( - 4 \beta + 2) q^{28} + (4 \beta + 2) q^{29} + ( - 2 \beta + 4) q^{30} + (\beta + 3) q^{32} - 2 \beta q^{33} + (6 \beta - 8) q^{34} + 2 q^{35} + ( - 10 \beta + 5) q^{36} + ( - 4 \beta + 2) q^{37} + (\beta - 3) q^{40} - 6 q^{41} + ( - 4 \beta + 8) q^{42} - 6 q^{43} + (2 \beta - 1) q^{44} + 5 q^{45} + ( - 2 \beta + 4) q^{46} + 2 \beta q^{47} + 6 \beta q^{48} - 3 q^{49} + (\beta - 1) q^{50} + (8 \beta - 8) q^{51} + ( - 4 \beta + 6) q^{53} + ( - 4 \beta + 8) q^{54} - q^{55} + (2 \beta - 6) q^{56} + ( - 2 \beta + 6) q^{58} + (4 \beta + 4) q^{59} + (2 \beta - 8) q^{60} + (8 \beta + 2) q^{61} + 10 q^{63} + (2 \beta - 7) q^{64} + (2 \beta - 4) q^{66} + (6 \beta - 4) q^{67} + ( - 10 \beta + 12) q^{68} + 8 q^{69} + (2 \beta - 2) q^{70} + 8 \beta q^{71} + (5 \beta - 15) q^{72} + (2 \beta + 4) q^{73} + (6 \beta - 10) q^{74} + 2 \beta q^{75} - 2 q^{77} + 4 q^{79} + 3 q^{80} + q^{81} + ( - 6 \beta + 6) q^{82} + 6 q^{83} + (4 \beta - 16) q^{84} + ( - 2 \beta + 4) q^{85} + ( - 6 \beta + 6) q^{86} + (4 \beta + 16) q^{87} + ( - \beta + 3) q^{88} + ( - 8 \beta + 2) q^{89} + (5 \beta - 5) q^{90} + (2 \beta - 8) q^{92} + ( - 2 \beta + 4) q^{94} + (6 \beta + 4) q^{96} + (4 \beta + 2) q^{97} + ( - 3 \beta + 3) q^{98} - 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9} - 2 q^{10} - 2 q^{11} - 16 q^{12} - 4 q^{14} + 6 q^{16} + 8 q^{17} - 10 q^{18} + 2 q^{20} + 2 q^{22} + 8 q^{24} + 2 q^{25} + 4 q^{28} + 4 q^{29} + 8 q^{30} + 6 q^{32} - 16 q^{34} + 4 q^{35} + 10 q^{36} + 4 q^{37} - 6 q^{40} - 12 q^{41} + 16 q^{42} - 12 q^{43} - 2 q^{44} + 10 q^{45} + 8 q^{46} - 6 q^{49} - 2 q^{50} - 16 q^{51} + 12 q^{53} + 16 q^{54} - 2 q^{55} - 12 q^{56} + 12 q^{58} + 8 q^{59} - 16 q^{60} + 4 q^{61} + 20 q^{63} - 14 q^{64} - 8 q^{66} - 8 q^{67} + 24 q^{68} + 16 q^{69} - 4 q^{70} - 30 q^{72} + 8 q^{73} - 20 q^{74} - 4 q^{77} + 8 q^{79} + 6 q^{80} + 2 q^{81} + 12 q^{82} + 12 q^{83} - 32 q^{84} + 8 q^{85} + 12 q^{86} + 32 q^{87} + 6 q^{88} + 4 q^{89} - 10 q^{90} - 16 q^{92} + 8 q^{94} + 8 q^{96} + 4 q^{97} + 6 q^{98} - 10 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
−2.41421 −2.82843 3.82843 1.00000 6.82843 2.00000 −4.41421 5.00000 −2.41421
1.2 0.414214 2.82843 −1.82843 1.00000 1.17157 2.00000 −1.58579 5.00000 0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

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 9295.2.a.g 2
13.b even 2 1 55.2.a.b 2
39.d odd 2 1 495.2.a.b 2
52.b odd 2 1 880.2.a.m 2
65.d even 2 1 275.2.a.c 2
65.h odd 4 2 275.2.b.d 4
91.b odd 2 1 2695.2.a.f 2
104.e even 2 1 3520.2.a.bn 2
104.h odd 2 1 3520.2.a.bo 2
143.d odd 2 1 605.2.a.d 2
143.l odd 10 4 605.2.g.l 8
143.n even 10 4 605.2.g.f 8
156.h even 2 1 7920.2.a.ch 2
195.e odd 2 1 2475.2.a.x 2
195.s even 4 2 2475.2.c.l 4
260.g odd 2 1 4400.2.a.bn 2
260.p even 4 2 4400.2.b.q 4
429.e even 2 1 5445.2.a.y 2
572.b even 2 1 9680.2.a.bn 2
715.c odd 2 1 3025.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 13.b even 2 1
275.2.a.c 2 65.d even 2 1
275.2.b.d 4 65.h odd 4 2
495.2.a.b 2 39.d odd 2 1
605.2.a.d 2 143.d odd 2 1
605.2.g.f 8 143.n even 10 4
605.2.g.l 8 143.l odd 10 4
880.2.a.m 2 52.b odd 2 1
2475.2.a.x 2 195.e odd 2 1
2475.2.c.l 4 195.s even 4 2
2695.2.a.f 2 91.b odd 2 1
3025.2.a.o 2 715.c odd 2 1
3520.2.a.bn 2 104.e even 2 1
3520.2.a.bo 2 104.h odd 2 1
4400.2.a.bn 2 260.g odd 2 1
4400.2.b.q 4 260.p even 4 2
5445.2.a.y 2 429.e even 2 1
7920.2.a.ch 2 156.h even 2 1
9295.2.a.g 2 1.a even 1 1 trivial
9680.2.a.bn 2 572.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

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