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Properties

Label 9295.2.a.t
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $0$
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: \(0\)
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: \( x^{9} - x^{8} - 16x^{7} + 14x^{6} + 86x^{5} - 57x^{4} - 179x^{3} + 64x^{2} + 118x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{8} - \beta_{4}) q^{6} + \beta_{7} q^{7} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{8} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{8} - \beta_{4}) q^{6} + \beta_{7} q^{7} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{8} + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+ \cdots + (\beta_{7} - \beta_{4} + \beta_{3} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{2} + 2 q^{3} + 15 q^{4} - 9 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{2} + 2 q^{3} + 15 q^{4} - 9 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} + 23 q^{9} + q^{10} + 9 q^{11} + 6 q^{12} + 16 q^{14} - 2 q^{15} + 15 q^{16} + 13 q^{17} - 3 q^{18} + 3 q^{19} - 15 q^{20} - 14 q^{21} - q^{22} - 2 q^{24} + 9 q^{25} + 8 q^{27} + 27 q^{28} + 16 q^{29} - q^{30} - 12 q^{31} + 18 q^{32} + 2 q^{33} - 14 q^{34} - 4 q^{35} - 4 q^{36} - q^{37} - 9 q^{38} + 3 q^{40} - 5 q^{41} - 11 q^{42} - 3 q^{43} + 15 q^{44} - 23 q^{45} - 4 q^{46} - 27 q^{47} + q^{48} + 31 q^{49} - q^{50} + 34 q^{53} + 49 q^{54} - 9 q^{55} + 73 q^{56} + 16 q^{57} + 38 q^{58} + 4 q^{59} - 6 q^{60} + 30 q^{61} - 8 q^{62} + 30 q^{63} + 3 q^{64} + q^{66} + 11 q^{67} + 52 q^{68} + 52 q^{69} - 16 q^{70} + 4 q^{71} + 8 q^{72} + 12 q^{73} - 16 q^{74} + 2 q^{75} + 16 q^{76} + 4 q^{77} - 16 q^{79} - 15 q^{80} + 61 q^{81} - 42 q^{82} - 34 q^{83} + 79 q^{84} - 13 q^{85} - 52 q^{86} - 64 q^{87} - 3 q^{88} + 10 q^{89} + 3 q^{90} + 7 q^{92} + 22 q^{93} - 18 q^{94} - 3 q^{95} + 15 q^{97} - 24 q^{98} + 23 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 16x^{7} + 14x^{6} + 86x^{5} - 57x^{4} - 179x^{3} + 64x^{2} + 118x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{8} + \nu^{7} + 20\nu^{6} - 18\nu^{5} - 31\nu^{4} + 86\nu^{3} - 99\nu^{2} - 115\nu + 94 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{8} + 4\nu^{7} + 105\nu^{6} - 72\nu^{5} - 399\nu^{4} + 344\nu^{3} + 379\nu^{2} - 385\nu - 24 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{8} - 9\nu^{7} + 95\nu^{6} + 112\nu^{5} - 371\nu^{4} - 399\nu^{3} + 341\nu^{2} + 335\nu - 21 ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{8} - 9\nu^{7} + 95\nu^{6} + 112\nu^{5} - 396\nu^{4} - 399\nu^{3} + 516\nu^{2} + 310\nu - 171 ) / 25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{8} - 9\nu^{7} + 95\nu^{6} + 112\nu^{5} - 396\nu^{4} - 424\nu^{3} + 516\nu^{2} + 435\nu - 146 ) / 25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{8} + 16\nu^{7} + 95\nu^{6} - 213\nu^{5} - 371\nu^{4} + 801\nu^{3} + 366\nu^{2} - 715\nu + 4 ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} + 7\beta_{2} - \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - 8\beta_{7} + 9\beta_{6} - 2\beta_{4} + \beta_{3} - \beta_{2} + 30\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{6} + 11\beta_{5} + \beta_{4} - 4\beta_{3} + 46\beta_{2} - 14\beta _1 + 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14\beta_{8} - 56\beta_{7} + 69\beta_{6} - \beta_{5} - 26\beta_{4} + 13\beta_{3} - 14\beta_{2} + 192\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{8} + \beta_{7} - 98\beta_{6} + 94\beta_{5} + 15\beta_{4} - 55\beta_{3} + 304\beta_{2} - 141\beta _1 + 852 \) 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
2.54892
2.34321
2.16069
1.13118
0.113915
−1.06779
−1.32285
−2.20826
−2.69902
−2.54892 −1.71826 4.49699 −1.00000 4.37972 −3.27678 −6.36464 −0.0475696 2.54892
1.2 −2.34321 2.48065 3.49065 −1.00000 −5.81268 3.89108 −3.49290 3.15360 2.34321
1.3 −2.16069 1.29393 2.66858 −1.00000 −2.79578 −2.72538 −1.44460 −1.32574 2.16069
1.4 −1.13118 −3.33877 −0.720430 −1.00000 3.77675 4.42443 3.07730 8.14739 1.13118
1.5 −0.113915 3.18947 −1.98702 −1.00000 −0.363329 −3.61768 0.454183 7.17269 0.113915
1.6 1.06779 0.343328 −0.859816 −1.00000 0.366604 −1.31980 −3.05369 −2.88213 −1.06779
1.7 1.32285 −2.67512 −0.250078 −1.00000 −3.53878 0.156885 −2.97651 4.15628 −1.32285
1.8 2.20826 3.17271 2.87640 −1.00000 7.00615 1.59773 1.93531 7.06608 −2.20826
1.9 2.69902 −0.747922 5.28473 −1.00000 −2.01866 4.86951 8.86555 −2.44061 −2.69902
\(n\): 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.t 9
13.b even 2 1 715.2.a.i ✓ 9
39.d odd 2 1 6435.2.a.bq 9
65.d even 2 1 3575.2.a.t 9
143.d odd 2 1 7865.2.a.w 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.a.i ✓ 9 13.b even 2 1
3575.2.a.t 9 65.d even 2 1
6435.2.a.bq 9 39.d odd 2 1
7865.2.a.w 9 143.d odd 2 1
9295.2.a.t 9 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{9} + T_{2}^{8} - 16T_{2}^{7} - 14T_{2}^{6} + 86T_{2}^{5} + 57T_{2}^{4} - 179T_{2}^{3} - 64T_{2}^{2} + 118T_{2} + 14 \) Copy content Toggle raw display
\( T_{3}^{9} - 2T_{3}^{8} - 23T_{3}^{7} + 42T_{3}^{6} + 169T_{3}^{5} - 260T_{3}^{4} - 434T_{3}^{3} + 466T_{3}^{2} + 272T_{3} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + T^{8} + \cdots + 14 \) Copy content Toggle raw display
$3$ \( T^{9} - 2 T^{8} + \cdots - 128 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 4 T^{8} + \cdots - 896 \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 13 T^{8} + \cdots - 3776 \) Copy content Toggle raw display
$19$ \( T^{9} - 3 T^{8} + \cdots - 1285360 \) Copy content Toggle raw display
$23$ \( T^{9} - 111 T^{7} + \cdots + 219520 \) Copy content Toggle raw display
$29$ \( T^{9} - 16 T^{8} + \cdots - 2929024 \) Copy content Toggle raw display
$31$ \( T^{9} + 12 T^{8} + \cdots - 131072 \) Copy content Toggle raw display
$37$ \( T^{9} + T^{8} + \cdots + 7082432 \) Copy content Toggle raw display
$41$ \( T^{9} + 5 T^{8} + \cdots + 155500 \) Copy content Toggle raw display
$43$ \( T^{9} + 3 T^{8} + \cdots - 256 \) Copy content Toggle raw display
$47$ \( T^{9} + 27 T^{8} + \cdots + 24153344 \) Copy content Toggle raw display
$53$ \( T^{9} - 34 T^{8} + \cdots + 283264 \) Copy content Toggle raw display
$59$ \( T^{9} - 4 T^{8} + \cdots + 17783296 \) Copy content Toggle raw display
$61$ \( T^{9} - 30 T^{8} + \cdots + 114329728 \) Copy content Toggle raw display
$67$ \( T^{9} - 11 T^{8} + \cdots - 1252352 \) Copy content Toggle raw display
$71$ \( T^{9} - 4 T^{8} + \cdots + 702464 \) Copy content Toggle raw display
$73$ \( T^{9} - 12 T^{8} + \cdots - 100765000 \) Copy content Toggle raw display
$79$ \( T^{9} + 16 T^{8} + \cdots + 49283072 \) Copy content Toggle raw display
$83$ \( T^{9} + 34 T^{8} + \cdots + 88272128 \) Copy content Toggle raw display
$89$ \( T^{9} - 10 T^{8} + \cdots + 560000 \) Copy content Toggle raw display
$97$ \( T^{9} - 15 T^{8} + \cdots - 1863872 \) Copy content Toggle raw display
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