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Properties

Label 9295.2.a.z
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 22 x^{12} + 19 x^{11} + 186 x^{10} - 134 x^{9} - 763 x^{8} + 440 x^{7} + 1573 x^{6} + \cdots - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{9} - \beta_{4} - \beta_1) q^{6} + \beta_{12} q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{7} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{9} - \beta_{4} - \beta_1) q^{6} + \beta_{12} q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{7} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{6} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} + q^{3} + 17 q^{4} - 14 q^{5} - 8 q^{6} - 3 q^{7} - 6 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} + q^{3} + 17 q^{4} - 14 q^{5} - 8 q^{6} - 3 q^{7} - 6 q^{8} + 19 q^{9} + q^{10} + 14 q^{11} + 23 q^{12} + 4 q^{14} - q^{15} + 23 q^{16} + 13 q^{17} + 3 q^{18} - q^{19} - 17 q^{20} - 6 q^{21} - q^{22} + 6 q^{23} + q^{24} + 14 q^{25} - 2 q^{27} - 25 q^{28} + 7 q^{29} + 8 q^{30} + 11 q^{31} - 18 q^{32} + q^{33} - 8 q^{34} + 3 q^{35} + 19 q^{36} - 2 q^{37} + 9 q^{38} + 6 q^{40} - 5 q^{41} + 4 q^{42} + 25 q^{43} + 17 q^{44} - 19 q^{45} + 11 q^{46} - 12 q^{47} + 24 q^{48} + 25 q^{49} - q^{50} + 36 q^{51} + 10 q^{53} - 14 q^{54} - 14 q^{55} + 16 q^{56} - 18 q^{57} + 20 q^{58} + 28 q^{59} - 23 q^{60} + 16 q^{61} + 31 q^{62} - 22 q^{63} + 50 q^{64} - 8 q^{66} + q^{67} + 4 q^{68} + 22 q^{69} - 4 q^{70} - 8 q^{71} + 35 q^{72} - 31 q^{73} + 23 q^{74} + q^{75} + 6 q^{76} - 3 q^{77} - 15 q^{79} - 23 q^{80} + 18 q^{81} + 12 q^{82} + 20 q^{83} - 72 q^{84} - 13 q^{85} - 10 q^{86} + 38 q^{87} - 6 q^{88} - 8 q^{89} - 3 q^{90} + 31 q^{92} + 33 q^{93} - 60 q^{94} + q^{95} + 36 q^{96} - 49 q^{97} - 12 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 22 x^{12} + 19 x^{11} + 186 x^{10} - 134 x^{9} - 763 x^{8} + 440 x^{7} + 1573 x^{6} + \cdots - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{13} - \nu^{12} - 46 \nu^{11} + 19 \nu^{10} + 403 \nu^{9} - 136 \nu^{8} - 1672 \nu^{7} + \cdots - 13 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{13} + 4 \nu^{12} + 71 \nu^{11} - 86 \nu^{10} - 649 \nu^{9} + 711 \nu^{8} + 2871 \nu^{7} + \cdots - 62 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{13} - 8 \nu^{12} - 189 \nu^{11} + 140 \nu^{10} + 1491 \nu^{9} - 885 \nu^{8} - 5505 \nu^{7} + \cdots - 12 ) / 26 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9 \nu^{13} + \nu^{12} - 187 \nu^{11} - 67 \nu^{10} + 1427 \nu^{9} + 922 \nu^{8} - 4830 \nu^{7} + \cdots + 147 ) / 26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{13} + 4 \nu^{12} - 195 \nu^{11} - 110 \nu^{10} + 1579 \nu^{9} + 1065 \nu^{8} - 5879 \nu^{7} + \cdots + 44 ) / 26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20 \nu^{13} - 11 \nu^{12} - 414 \nu^{11} + 161 \nu^{10} + 3182 \nu^{9} - 697 \nu^{8} - 11179 \nu^{7} + \cdots + 95 ) / 26 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16 \nu^{13} + \nu^{12} + 352 \nu^{11} + 9 \nu^{10} - 2920 \nu^{9} - 277 \nu^{8} + 11317 \nu^{7} + \cdots - 89 ) / 26 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{13} + 23 \nu^{12} + 55 \nu^{11} - 471 \nu^{10} - 397 \nu^{9} + 3584 \nu^{8} + 1592 \nu^{7} + \cdots + 83 ) / 26 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12 \nu^{13} - 19 \nu^{12} - 250 \nu^{11} + 361 \nu^{10} + 1976 \nu^{9} - 2519 \nu^{8} - 7471 \nu^{7} + \cdots + 13 ) / 26 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 24 \nu^{13} + \nu^{12} + 512 \nu^{11} + 51 \nu^{10} - 4076 \nu^{9} - 955 \nu^{8} + 14897 \nu^{7} + \cdots + 83 ) / 26 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10 \nu^{13} - 4 \nu^{12} + 215 \nu^{11} + 112 \nu^{10} - 1717 \nu^{9} - 1088 \nu^{8} + 6229 \nu^{7} + \cdots - 16 ) / 13 \) 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
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{7} + \beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{10} + \beta_{8} - \beta_{4} + 8\beta_{2} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 9 \beta_{11} + 10 \beta_{10} - \beta_{9} - 10 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{13} - \beta_{12} + 12 \beta_{10} + \beta_{9} + 11 \beta_{8} - \beta_{6} + 2 \beta_{5} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 17 \beta_{13} - 2 \beta_{12} + 68 \beta_{11} + 84 \beta_{10} - 15 \beta_{9} + \beta_{8} - 84 \beta_{7} + \cdots + 68 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 113 \beta_{13} - 14 \beta_{12} + 3 \beta_{11} + 116 \beta_{10} + 14 \beta_{9} + 98 \beta_{8} + \cdots + 659 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 195 \beta_{13} - 32 \beta_{12} + 497 \beta_{11} + 675 \beta_{10} - 158 \beta_{9} + 19 \beta_{8} + \cdots + 495 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 978 \beta_{13} - 142 \beta_{12} + 58 \beta_{11} + 1036 \beta_{10} + 136 \beta_{9} + 816 \beta_{8} + \cdots + 4562 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1905 \beta_{13} - 354 \beta_{12} + 3625 \beta_{11} + 5337 \beta_{10} - 1442 \beta_{9} + 245 \beta_{8} + \cdots + 3633 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8132 \beta_{13} - 1277 \beta_{12} + 745 \beta_{11} + 8874 \beta_{10} + 1132 \beta_{9} + 6582 \beta_{8} + \cdots + 32355 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17136 \beta_{13} - 3372 \beta_{12} + 26585 \beta_{11} + 41870 \beta_{10} - 12227 \beta_{9} + \cdots + 27264 \) 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
2.79391
2.39349
2.35360
1.38616
1.28865
1.24204
0.0358458
−0.254073
−0.357005
−1.49038
−1.61565
−1.73250
−2.35860
−2.68547
−2.79391 0.603677 5.80593 −1.00000 −1.68662 −3.21310 −10.6334 −2.63557 2.79391
1.2 −2.39349 −0.884071 3.72879 −1.00000 2.11601 0.827927 −4.13783 −2.21842 2.39349
1.3 −2.35360 3.38932 3.53942 −1.00000 −7.97709 −0.359069 −3.62317 8.48747 2.35360
1.4 −1.38616 1.70116 −0.0785728 −1.00000 −2.35808 2.83341 2.88123 −0.106045 1.38616
1.5 −1.28865 1.74443 −0.339389 −1.00000 −2.24795 −4.84380 3.01465 0.0430292 1.28865
1.6 −1.24204 −2.93660 −0.457342 −1.00000 3.64736 2.02860 3.05211 5.62360 1.24204
1.7 −0.0358458 0.771180 −1.99872 −1.00000 −0.0276436 4.74642 0.143337 −2.40528 0.0358458
1.8 0.254073 −1.28795 −1.93545 −1.00000 −0.327235 −0.181090 −0.999892 −1.34117 −0.254073
1.9 0.357005 −3.05288 −1.87255 −1.00000 −1.08989 −5.21395 −1.38252 6.32007 −0.357005
1.10 1.49038 2.63623 0.221247 −1.00000 3.92900 −0.108508 −2.65103 3.94970 −1.49038
1.11 1.61565 0.304356 0.610311 −1.00000 0.491731 −1.14764 −2.24525 −2.90737 −1.61565
1.12 1.73250 −2.30813 1.00155 −1.00000 −3.99883 4.56389 −1.72981 2.32747 −1.73250
1.13 2.35860 −2.05525 3.56299 −1.00000 −4.84752 −1.30895 3.68648 1.22407 −2.35860
1.14 2.68547 2.37454 5.21177 −1.00000 6.37676 −1.62414 8.62514 2.63843 −2.68547
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.z 14
13.b even 2 1 9295.2.a.bc 14
13.e even 6 2 715.2.i.d ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.i.d ✓ 28 13.e even 6 2
9295.2.a.z 14 1.a even 1 1 trivial
9295.2.a.bc 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):

\( T_{2}^{14} + T_{2}^{13} - 22 T_{2}^{12} - 19 T_{2}^{11} + 186 T_{2}^{10} + 134 T_{2}^{9} - 763 T_{2}^{8} + \cdots - 3 \) Copy content Toggle raw display
\( T_{3}^{14} - T_{3}^{13} - 30 T_{3}^{12} + 29 T_{3}^{11} + 341 T_{3}^{10} - 328 T_{3}^{9} - 1839 T_{3}^{8} + \cdots + 432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{14} - T^{13} + \cdots + 432 \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 3 T^{13} + \cdots - 144 \) Copy content Toggle raw display
$11$ \( (T - 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} \) Copy content Toggle raw display
$17$ \( T^{14} - 13 T^{13} + \cdots - 3005181 \) Copy content Toggle raw display
$19$ \( T^{14} + T^{13} + \cdots + 25965424 \) Copy content Toggle raw display
$23$ \( T^{14} - 6 T^{13} + \cdots - 32111616 \) Copy content Toggle raw display
$29$ \( T^{14} - 7 T^{13} + \cdots + 49071168 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 128151641 \) Copy content Toggle raw display
$37$ \( T^{14} + 2 T^{13} + \cdots - 17786368 \) Copy content Toggle raw display
$41$ \( T^{14} + 5 T^{13} + \cdots + 543312 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 19632303367 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 1512346608 \) Copy content Toggle raw display
$53$ \( T^{14} - 10 T^{13} + \cdots - 54636528 \) Copy content Toggle raw display
$59$ \( T^{14} - 28 T^{13} + \cdots - 760443 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 21301035008 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 9576262352 \) Copy content Toggle raw display
$71$ \( T^{14} + 8 T^{13} + \cdots + 29081856 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 260532099367 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 98946764608 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 125474483307 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 18234711052203 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 543202592512 \) Copy content Toggle raw display
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