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Properties

Label 9295.2.a.s
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 11x^{7} + 9x^{6} + 35x^{5} - 18x^{4} - 40x^{3} + 4x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 715)
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 a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{5} - \beta_{3} + 1) q^{6} + (\beta_{8} + \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{5} - \beta_{3} + 1) q^{6} + (\beta_{8} + \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 1) q^{8}+ \cdots + (\beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{2} + 5 q^{4} + 9 q^{5} + 5 q^{6} - 4 q^{7} - 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{2} + 5 q^{4} + 9 q^{5} + 5 q^{6} - 4 q^{7} - 3 q^{8} - q^{9} - q^{10} - 9 q^{11} - 6 q^{12} - 12 q^{14} + 9 q^{16} - 13 q^{17} - 15 q^{18} - 5 q^{19} + 5 q^{20} - 10 q^{21} + q^{22} - 2 q^{23} + 14 q^{24} + 9 q^{25} + 11 q^{28} - 12 q^{29} + 5 q^{30} + 14 q^{31} + 8 q^{32} + 8 q^{34} - 4 q^{35} - 18 q^{36} - 5 q^{37} - 3 q^{38} - 3 q^{40} + 5 q^{41} + 27 q^{42} + 5 q^{43} - 5 q^{44} - q^{45} - 6 q^{46} - 11 q^{47} - 11 q^{48} + 11 q^{49} - q^{50} - 26 q^{53} - 7 q^{54} - 9 q^{55} - 19 q^{56} + 8 q^{57} - 26 q^{58} + 12 q^{59} - 6 q^{60} + 6 q^{61} - 22 q^{62} + 22 q^{63} + 17 q^{64} - 5 q^{66} + 9 q^{67} + 16 q^{68} - 12 q^{69} - 12 q^{70} + 10 q^{71} + 8 q^{72} - 10 q^{73} - 32 q^{74} + 12 q^{76} + 4 q^{77} - 28 q^{79} + 9 q^{80} - 11 q^{81} - 32 q^{82} + 12 q^{83} - 29 q^{84} - 13 q^{85} - 18 q^{86} - 16 q^{87} + 3 q^{88} - 12 q^{89} - 15 q^{90} - 41 q^{92} + 2 q^{93} + 66 q^{94} - 5 q^{95} - 20 q^{96} - 3 q^{97} - 20 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 11x^{7} + 9x^{6} + 35x^{5} - 18x^{4} - 40x^{3} + 4x^{2} + 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{8} - 3\nu^{7} - 20\nu^{6} + 28\nu^{5} + 51\nu^{4} - 62\nu^{3} - 37\nu^{2} + 29\nu + 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 3\nu^{8} - 4\nu^{7} - 31\nu^{6} + 37\nu^{5} + 86\nu^{4} - 80\nu^{3} - 76\nu^{2} + 35\nu + 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 3\nu^{8} - 5\nu^{7} - 30\nu^{6} + 47\nu^{5} + 77\nu^{4} - 105\nu^{3} - 58\nu^{2} + 49\nu + 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 5\nu^{8} - 8\nu^{7} - 51\nu^{6} + 76\nu^{5} + 138\nu^{4} - 175\nu^{3} - 120\nu^{2} + 90\nu + 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -6\nu^{8} + 9\nu^{7} + 62\nu^{6} - 85\nu^{5} - 173\nu^{4} + 194\nu^{3} + 159\nu^{2} - 100\nu - 57 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 8\nu^{8} - 11\nu^{7} - 83\nu^{6} + 103\nu^{5} + 232\nu^{4} - 231\nu^{3} - 207\nu^{2} + 115\nu + 67 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + 10\beta_{6} - 2\beta_{5} + 8\beta_{4} - 7\beta_{3} + 2\beta_{2} + 21\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{8} - 9\beta_{7} + \beta_{6} - 11\beta_{5} + 10\beta_{4} + 12\beta_{3} + 47\beta_{2} + \beta _1 + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{8} + 65\beta_{7} + 76\beta_{6} - 23\beta_{5} + 57\beta_{4} - 42\beta_{3} + 22\beta_{2} + 125\beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -76\beta_{8} - 62\beta_{7} + 15\beta_{6} - 91\beta_{5} + 79\beta_{4} + 99\beta_{3} + 315\beta_{2} + 13\beta _1 + 552 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.60808
1.89467
1.34647
0.941231
−0.424633
−0.497664
−0.754950
−1.58115
−2.53205
−2.60808 −0.816557 4.80207 1.00000 2.12964 0.643124 −7.30803 −2.33323 −2.60808
1.2 −1.89467 −2.72571 1.58978 1.00000 5.16433 4.14336 0.777224 4.42949 −1.89467
1.3 −1.34647 2.52386 −0.187030 1.00000 −3.39829 0.232088 2.94476 3.36985 −1.34647
1.4 −0.941231 0.891896 −1.11408 1.00000 −0.839481 −4.41096 2.93107 −2.20452 −0.941231
1.5 0.424633 −1.07487 −1.81969 1.00000 −0.456426 2.74315 −1.62196 −1.84465 0.424633
1.6 0.497664 −1.95418 −1.75233 1.00000 −0.972525 −5.06031 −1.86740 0.818821 0.497664
1.7 0.754950 2.32364 −1.43005 1.00000 1.75423 −0.530575 −2.58952 2.39930 0.754950
1.8 1.58115 0.513164 0.500029 1.00000 0.811388 0.106663 −2.37168 −2.73666 1.58115
1.9 2.53205 0.318764 4.41129 1.00000 0.807127 −1.86655 6.10553 −2.89839 2.53205
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.s 9
13.b even 2 1 9295.2.a.u 9
13.d odd 4 2 715.2.e.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.e.b ✓ 18 13.d odd 4 2
9295.2.a.s 9 1.a even 1 1 trivial
9295.2.a.u 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):

\( T_{2}^{9} + T_{2}^{8} - 11T_{2}^{7} - 9T_{2}^{6} + 35T_{2}^{5} + 18T_{2}^{4} - 40T_{2}^{3} - 4T_{2}^{2} + 16T_{2} - 4 \) Copy content Toggle raw display
\( T_{3}^{9} - 13T_{3}^{7} + 49T_{3}^{5} + 2T_{3}^{4} - 52T_{3}^{3} + 4T_{3}^{2} + 16T_{3} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + T^{8} - 11 T^{7} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( T^{9} - 13 T^{7} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 4 T^{8} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 13 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{9} + 5 T^{8} + \cdots + 2411 \) Copy content Toggle raw display
$23$ \( T^{9} + 2 T^{8} + \cdots - 130048 \) Copy content Toggle raw display
$29$ \( T^{9} + 12 T^{8} + \cdots + 279232 \) Copy content Toggle raw display
$31$ \( T^{9} - 14 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{9} + 5 T^{8} + \cdots + 339088 \) Copy content Toggle raw display
$41$ \( T^{9} - 5 T^{8} + \cdots + 105109 \) Copy content Toggle raw display
$43$ \( T^{9} - 5 T^{8} + \cdots + 19792 \) Copy content Toggle raw display
$47$ \( T^{9} + 11 T^{8} + \cdots - 1673968 \) Copy content Toggle raw display
$53$ \( T^{9} + 26 T^{8} + \cdots - 174816 \) Copy content Toggle raw display
$59$ \( T^{9} - 12 T^{8} + \cdots - 6364224 \) Copy content Toggle raw display
$61$ \( T^{9} - 6 T^{8} + \cdots + 369088 \) Copy content Toggle raw display
$67$ \( T^{9} - 9 T^{8} + \cdots - 226704 \) Copy content Toggle raw display
$71$ \( T^{9} - 10 T^{8} + \cdots + 167958464 \) Copy content Toggle raw display
$73$ \( T^{9} + 10 T^{8} + \cdots + 95308 \) Copy content Toggle raw display
$79$ \( T^{9} + 28 T^{8} + \cdots - 1395712 \) Copy content Toggle raw display
$83$ \( T^{9} - 12 T^{8} + \cdots - 12752064 \) Copy content Toggle raw display
$89$ \( T^{9} + 12 T^{8} + \cdots - 98431296 \) Copy content Toggle raw display
$97$ \( T^{9} + 3 T^{8} + \cdots + 22214912 \) Copy content Toggle raw display
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