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Properties

Label 9295.2.a.bf
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $1$
Dimension $27$
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: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 10 q^{2} - 12 q^{3} + 20 q^{4} + 27 q^{5} + 10 q^{6} - 8 q^{7} - 33 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 10 q^{2} - 12 q^{3} + 20 q^{4} + 27 q^{5} + 10 q^{6} - 8 q^{7} - 33 q^{8} + 23 q^{9} - 10 q^{10} + 27 q^{11} - 34 q^{12} - 7 q^{14} - 12 q^{15} + 10 q^{16} - 5 q^{17} - 26 q^{18} - 6 q^{19} + 20 q^{20} - 31 q^{21} - 10 q^{22} - 13 q^{23} + 60 q^{24} + 27 q^{25} - 42 q^{27} - q^{28} + q^{29} + 10 q^{30} - 5 q^{31} - 21 q^{32} - 12 q^{33} + 21 q^{34} - 8 q^{35} + 18 q^{36} - 24 q^{37} + 12 q^{38} - 33 q^{40} - 66 q^{41} + 5 q^{42} + 2 q^{43} + 20 q^{44} + 23 q^{45} - 31 q^{46} - 68 q^{47} - 74 q^{48} + 9 q^{49} - 10 q^{50} - 7 q^{51} - 25 q^{53} + 22 q^{54} + 27 q^{55} - 36 q^{57} - 21 q^{58} - 58 q^{59} - 34 q^{60} + 6 q^{61} + 4 q^{62} + 68 q^{63} + 13 q^{64} + 10 q^{66} - 40 q^{67} + 20 q^{68} + 36 q^{69} - 7 q^{70} - 35 q^{71} - 96 q^{72} - 11 q^{73} + 22 q^{74} - 12 q^{75} - 26 q^{76} - 8 q^{77} + 22 q^{79} + 10 q^{80} + 31 q^{81} + 44 q^{82} - 6 q^{83} - 3 q^{84} - 5 q^{85} - 71 q^{86} - 22 q^{87} - 33 q^{88} - 77 q^{89} - 26 q^{90} + 5 q^{92} - 74 q^{93} + 64 q^{94} - 6 q^{95} + 91 q^{96} - 42 q^{97} - 36 q^{98} + 23 q^{99}+O(q^{100}) \) 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 −2.73241 −3.29392 5.46608 1.00000 9.00035 4.07731 −9.47076 7.84991 −2.73241
1.2 −2.66935 −1.68397 5.12545 1.00000 4.49512 −3.30182 −8.34294 −0.164240 −2.66935
1.3 −2.57525 0.0709162 4.63190 1.00000 −0.182627 −2.32233 −6.77780 −2.99497 −2.57525
1.4 −2.34300 −2.51564 3.48964 1.00000 5.89414 −3.74092 −3.49022 3.32845 −2.34300
1.5 −2.23119 2.20686 2.97822 1.00000 −4.92394 2.19989 −2.18259 1.87025 −2.23119
1.6 −2.10606 −0.0290764 2.43548 1.00000 0.0612366 1.49789 −0.917152 −2.99915 −2.10606
1.7 −2.03670 −1.88687 2.14815 1.00000 3.84298 3.18602 −0.301734 0.560262 −2.03670
1.8 −1.82838 −2.42349 1.34296 1.00000 4.43105 0.706642 1.20131 2.87330 −1.82838
1.9 −1.58114 2.46262 0.500006 1.00000 −3.89375 −1.56659 2.37170 3.06449 −1.58114
1.10 −1.49769 0.390668 0.243070 1.00000 −0.585098 2.22730 2.63133 −2.84738 −1.49769
1.11 −1.05535 1.49425 −0.886238 1.00000 −1.57696 −5.21667 3.04599 −0.767214 −1.05535
1.12 −0.898106 2.54273 −1.19341 1.00000 −2.28364 −1.10506 2.86802 3.46548 −0.898106
1.13 −0.824281 −2.41288 −1.32056 1.00000 1.98889 0.250206 2.73707 2.82198 −0.824281
1.14 −0.0988389 −0.367139 −1.99023 1.00000 0.0362876 0.551980 0.394390 −2.86521 −0.0988389
1.15 0.0379575 −3.37676 −1.99856 1.00000 −0.128173 2.89038 −0.151775 8.40252 0.0379575
1.16 0.0662568 −0.0239073 −1.99561 1.00000 −0.00158402 −3.55608 −0.264736 −2.99943 0.0662568
1.17 0.179832 −0.250363 −1.96766 1.00000 −0.0450233 1.11122 −0.713513 −2.93732 0.179832
1.18 0.704008 1.65227 −1.50437 1.00000 1.16321 3.83866 −2.46711 −0.270005 0.704008
1.19 0.747301 −1.09830 −1.44154 1.00000 −0.820758 −2.82694 −2.57187 −1.79375 0.747301
1.20 1.02408 −2.83710 −0.951268 1.00000 −2.90541 −2.10981 −3.02232 5.04915 1.02408
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
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.bf ✓ 27
13.b even 2 1 9295.2.a.bi yes 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9295.2.a.bf ✓ 27 1.a even 1 1 trivial
9295.2.a.bi yes 27 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):

\( T_{2}^{27} + 10 T_{2}^{26} + 13 T_{2}^{25} - 179 T_{2}^{24} - 592 T_{2}^{23} + 1060 T_{2}^{22} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{27} + 12 T_{3}^{26} + 20 T_{3}^{25} - 298 T_{3}^{24} - 1253 T_{3}^{23} + 2152 T_{3}^{22} + \cdots + 1 \) Copy content Toggle raw display