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Properties

Label 9295.2.a.bj
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $0$
Dimension $28$
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: \(28\)
Twist minimal: no (minimal twist has level 715)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} + 6 q^{3} + 34 q^{4} - 28 q^{5} + 10 q^{6} - 8 q^{7} - 6 q^{8} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{2} + 6 q^{3} + 34 q^{4} - 28 q^{5} + 10 q^{6} - 8 q^{7} - 6 q^{8} + 46 q^{9} + 2 q^{10} - 28 q^{11} + 22 q^{12} + 16 q^{14} - 6 q^{15} + 54 q^{16} + 26 q^{17} - 36 q^{18} + 8 q^{19} - 34 q^{20} + 10 q^{21} + 2 q^{22} + 30 q^{23} + 26 q^{24} + 28 q^{25} + 4 q^{28} + 36 q^{29} - 10 q^{30} - 20 q^{31} + 16 q^{32} - 6 q^{33} - 8 q^{34} + 8 q^{35} + 58 q^{36} - 50 q^{37} + 18 q^{38} + 6 q^{40} + 6 q^{41} + 74 q^{42} - 34 q^{44} - 46 q^{45} + 24 q^{46} - 10 q^{47} + 50 q^{48} + 46 q^{49} - 2 q^{50} + 12 q^{51} + 52 q^{53} + 38 q^{54} + 28 q^{55} + 10 q^{57} - 42 q^{58} - 12 q^{59} - 22 q^{60} + 4 q^{61} + 14 q^{62} + 48 q^{63} + 56 q^{64} - 10 q^{66} - 52 q^{67} + 92 q^{68} - 16 q^{69} - 16 q^{70} + 20 q^{71} - 50 q^{72} - 40 q^{73} + 32 q^{74} + 6 q^{75} - 16 q^{76} + 8 q^{77} - 6 q^{79} - 54 q^{80} + 116 q^{81} - 36 q^{82} - 22 q^{83} + 86 q^{84} - 26 q^{85} + 32 q^{86} + 16 q^{87} + 6 q^{88} + 8 q^{89} + 36 q^{90} + 18 q^{92} - 40 q^{93} + 66 q^{94} - 8 q^{95} + 4 q^{96} - 48 q^{97} + 70 q^{98} - 46 q^{99}+O(q^{100}) \) 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 −2.67128 0.665607 5.13576 −1.00000 −1.77802 −3.49162 −8.37649 −2.55697 2.67128
1.2 −2.62861 −3.20847 4.90961 −1.00000 8.43383 1.70284 −7.64823 7.29429 2.62861
1.3 −2.60173 0.900904 4.76900 −1.00000 −2.34391 −0.351999 −7.20420 −2.18837 2.60173
1.4 −2.45496 3.15091 4.02682 −1.00000 −7.73535 −1.16506 −4.97577 6.92823 2.45496
1.5 −2.34794 2.51849 3.51284 −1.00000 −5.91328 3.00050 −3.55205 3.34280 2.34794
1.6 −1.98824 −3.25437 1.95309 −1.00000 6.47046 −2.47198 0.0932592 7.59091 1.98824
1.7 −1.97807 −2.13412 1.91278 −1.00000 4.22145 2.96984 0.172536 1.55448 1.97807
1.8 −1.70002 −1.13290 0.890055 −1.00000 1.92596 −0.836106 1.88692 −1.71653 1.70002
1.9 −1.33075 2.40091 −0.229109 −1.00000 −3.19500 −2.75056 2.96638 2.76436 1.33075
1.10 −1.33024 1.51613 −0.230448 −1.00000 −2.01682 −3.99670 2.96704 −0.701349 1.33024
1.11 −0.656450 3.35101 −1.56907 −1.00000 −2.19977 1.16416 2.34292 8.22925 0.656450
1.12 −0.655924 −1.34520 −1.56976 −1.00000 0.882352 −3.79230 2.34149 −1.19043 0.655924
1.13 −0.476015 −2.09754 −1.77341 −1.00000 0.998461 −4.49531 1.79620 1.39968 0.476015
1.14 −0.329478 −2.14759 −1.89144 −1.00000 0.707584 4.29216 1.28214 1.61215 0.329478
1.15 0.114343 1.96612 −1.98693 −1.00000 0.224812 1.17839 −0.455877 0.865622 −0.114343
1.16 0.479002 0.703661 −1.77056 −1.00000 0.337055 1.56312 −1.80610 −2.50486 −0.479002
1.17 0.594144 2.92222 −1.64699 −1.00000 1.73622 −3.48285 −2.16684 5.53938 −0.594144
1.18 0.821441 −2.95162 −1.32523 −1.00000 −2.42458 3.78321 −2.73148 5.71204 −0.821441
1.19 0.845362 0.534292 −1.28536 −1.00000 0.451670 0.778649 −2.77732 −2.71453 −0.845362
1.20 0.934466 −0.693282 −1.12677 −1.00000 −0.647848 0.485111 −2.92186 −2.51936 −0.934466
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
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.bj 28
13.b even 2 1 9295.2.a.bk 28
13.f odd 12 2 715.2.z.c ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.z.c ✓ 56 13.f odd 12 2
9295.2.a.bj 28 1.a even 1 1 trivial
9295.2.a.bk 28 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):

\( T_{2}^{28} + 2 T_{2}^{27} - 43 T_{2}^{26} - 84 T_{2}^{25} + 812 T_{2}^{24} + 1538 T_{2}^{23} + \cdots + 729 \) Copy content Toggle raw display
\( T_{3}^{28} - 6 T_{3}^{27} - 47 T_{3}^{26} + 342 T_{3}^{25} + 810 T_{3}^{24} - 8350 T_{3}^{23} + \cdots - 64256 \) Copy content Toggle raw display