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Properties

Label 9295.2.a.bl
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $1$
Dimension $33$
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: \(1\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 33 q - 10 q^{2} + 38 q^{4} + 33 q^{5} - 24 q^{6} - 6 q^{7} - 33 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q - 10 q^{2} + 38 q^{4} + 33 q^{5} - 24 q^{6} - 6 q^{7} - 33 q^{8} + 35 q^{9} - 10 q^{10} - 33 q^{11} - 12 q^{12} - q^{14} + 52 q^{16} - 21 q^{17} - 36 q^{18} - 18 q^{19} + 38 q^{20} - 33 q^{21} + 10 q^{22} + 11 q^{23} - 22 q^{24} + 33 q^{25} - 6 q^{27} - 23 q^{28} - 5 q^{29} - 24 q^{30} - 7 q^{31} - 121 q^{32} + 11 q^{34} - 6 q^{35} + 90 q^{36} - 18 q^{37} + 14 q^{38} - 33 q^{40} - 68 q^{41} + 23 q^{42} - 40 q^{43} - 38 q^{44} + 35 q^{45} - 11 q^{46} - 10 q^{47} + 49 q^{49} - 10 q^{50} - 3 q^{51} - 27 q^{53} + 14 q^{54} - 33 q^{55} + 2 q^{56} - 92 q^{57} + 5 q^{58} - 26 q^{59} - 12 q^{60} - 8 q^{61} + 2 q^{62} - 70 q^{63} + 115 q^{64} + 24 q^{66} - 14 q^{67} - 54 q^{68} - 60 q^{69} - q^{70} - 49 q^{71} - 108 q^{72} - 7 q^{73} + 30 q^{74} - 14 q^{76} + 6 q^{77} + 26 q^{79} + 52 q^{80} + 9 q^{81} - 56 q^{82} - 28 q^{83} - 17 q^{84} - 21 q^{85} - 83 q^{86} + 12 q^{87} + 33 q^{88} - 121 q^{89} - 36 q^{90} + 45 q^{92} - 6 q^{93} - 14 q^{94} - 18 q^{95} - 49 q^{96} - 52 q^{97} - 68 q^{98} - 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.81824 2.40306 5.94250 1.00000 −6.77240 −3.59423 −11.1109 2.77469 −2.81824
1.2 −2.81256 −2.99156 5.91047 1.00000 8.41394 −2.42306 −10.9984 5.94944 −2.81256
1.3 −2.81179 0.0592288 5.90616 1.00000 −0.166539 1.97880 −10.9833 −2.99649 −2.81179
1.4 −2.55877 0.257678 4.54729 1.00000 −0.659338 5.05687 −6.51792 −2.93360 −2.55877
1.5 −2.53031 3.21136 4.40248 1.00000 −8.12573 −1.13262 −6.07901 7.31281 −2.53031
1.6 −2.37279 2.60596 3.63011 1.00000 −6.18340 −2.57413 −3.86791 3.79105 −2.37279
1.7 −2.31895 −3.17790 3.37754 1.00000 7.36940 2.58351 −3.19445 7.09906 −2.31895
1.8 −1.83400 0.366540 1.36354 1.00000 −0.672233 −4.69603 1.16726 −2.86565 −1.83400
1.9 −1.81035 1.52166 1.27738 1.00000 −2.75474 −0.994042 1.30819 −0.684558 −1.81035
1.10 −1.64066 −2.66647 0.691766 1.00000 4.37478 −3.02590 2.14637 4.11008 −1.64066
1.11 −1.58120 −1.05603 0.500201 1.00000 1.66980 4.25529 2.37149 −1.88480 −1.58120
1.12 −1.31800 2.23086 −0.262864 1.00000 −2.94028 4.98346 2.98247 1.97671 −1.31800
1.13 −1.29701 1.34768 −0.317753 1.00000 −1.74797 0.856873 3.00616 −1.18375 −1.29701
1.14 −0.903753 −0.544099 −1.18323 1.00000 0.491731 −3.45551 2.87685 −2.70396 −0.903753
1.15 −0.856376 1.26783 −1.26662 1.00000 −1.08574 0.172869 2.79746 −1.39261 −0.856376
1.16 −0.537133 3.13453 −1.71149 1.00000 −1.68366 −1.28456 1.99356 6.82526 −0.537133
1.17 −0.463489 −1.52860 −1.78518 1.00000 0.708491 0.287666 1.75439 −0.663374 −0.463489
1.18 −0.0820056 2.99572 −1.99328 1.00000 −0.245666 −5.02925 0.327471 5.97436 −0.0820056
1.19 −0.0722894 −2.43942 −1.99477 1.00000 0.176345 1.57489 0.288780 2.95079 −0.0722894
1.20 −0.0614781 0.307609 −1.99622 1.00000 −0.0189112 3.20872 0.245680 −2.90538 −0.0614781
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
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.bl ✓ 33
13.b even 2 1 9295.2.a.bo yes 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9295.2.a.bl ✓ 33 1.a even 1 1 trivial
9295.2.a.bo yes 33 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):

\( T_{2}^{33} + 10 T_{2}^{32} - 2 T_{2}^{31} - 329 T_{2}^{30} - 724 T_{2}^{29} + 4395 T_{2}^{28} + \cdots - 49 \) Copy content Toggle raw display
\( T_{3}^{33} - 67 T_{3}^{31} + 2 T_{3}^{30} + 2015 T_{3}^{29} - 126 T_{3}^{28} - 35994 T_{3}^{27} + \cdots + 76531 \) Copy content Toggle raw display