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Properties

Label 9405.2.a.bq
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(1\)
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: \( x^{14} - x^{13} - 19 x^{12} + 15 x^{11} + 137 x^{10} - 80 x^{9} - 467 x^{8} + 193 x^{7} + 766 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} - \beta_1 q^{10} + q^{11} + ( - \beta_{11} + \beta_{6} - \beta_{2} - 1) q^{13} + (\beta_{9} + \beta_{6} + \beta_{5} + \cdots + 1) q^{14}+ \cdots + ( - \beta_{13} + \beta_{12} - \beta_{11} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 19 x^{12} + 15 x^{11} + 137 x^{10} - 80 x^{9} - 467 x^{8} + 193 x^{7} + 766 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 87 \nu^{13} + 48 \nu^{12} + 1753 \nu^{11} - 735 \nu^{10} - 13308 \nu^{9} + 4271 \nu^{8} + \cdots + 2295 ) / 569 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 234 \nu^{13} + 31 \nu^{12} + 4558 \nu^{11} + 201 \nu^{10} - 33204 \nu^{9} - 6073 \nu^{8} + \cdots + 2229 ) / 569 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 335 \nu^{13} + 538 \nu^{12} + 6253 \nu^{11} - 8736 \nu^{10} - 44749 \nu^{9} + 51593 \nu^{8} + \cdots + 5737 ) / 569 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 494 \nu^{13} + 61 \nu^{12} - 9496 \nu^{11} - 2890 \nu^{10} + 67442 \nu^{9} + 30523 \nu^{8} + \cdots - 533 ) / 569 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 529 \nu^{13} - 233 \nu^{12} - 9920 \nu^{11} + 2252 \nu^{10} + 69009 \nu^{9} - 2536 \nu^{8} + \cdots - 3281 ) / 569 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 685 \nu^{13} - 633 \nu^{12} - 12769 \nu^{11} + 8946 \nu^{10} + 89438 \nu^{9} - 43059 \nu^{8} + \cdots - 7043 ) / 569 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 747 \nu^{13} + 471 \nu^{12} + 13894 \nu^{11} - 5683 \nu^{10} - 96018 \nu^{9} + 19327 \nu^{8} + \cdots + 3067 ) / 569 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 818 \nu^{13} + 726 \nu^{12} + 15632 \nu^{11} - 10619 \nu^{10} - 113373 \nu^{9} + 54428 \nu^{8} + \cdots + 12592 ) / 569 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 818 \nu^{13} + 726 \nu^{12} + 15632 \nu^{11} - 10619 \nu^{10} - 113373 \nu^{9} + 54428 \nu^{8} + \cdots + 9178 ) / 569 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1157 \nu^{13} - 1070 \nu^{12} - 22031 \nu^{11} + 15602 \nu^{10} + 159244 \nu^{9} - 79252 \nu^{8} + \cdots - 15099 ) / 569 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{11} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} + \beta_{7} - 2\beta_{6} + 8\beta_{3} + 2\beta_{2} + 29\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} - 7 \beta_{12} + 10 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} + \beta_{7} - 4 \beta_{6} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{13} + 12 \beta_{12} + 14 \beta_{11} + \beta_{10} - 11 \beta_{9} - \beta_{8} + 14 \beta_{7} + \cdots + 91 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{13} - 38 \beta_{12} + 82 \beta_{11} - 14 \beta_{10} - 28 \beta_{9} - 16 \beta_{8} + 17 \beta_{7} + \cdots + 525 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 108 \beta_{13} + 106 \beta_{12} + 140 \beta_{11} + 13 \beta_{10} - 97 \beta_{9} - 15 \beta_{8} + \cdots + 688 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 155 \beta_{13} - 180 \beta_{12} + 632 \beta_{11} - 135 \beta_{10} - 280 \beta_{9} - 167 \beta_{8} + \cdots + 3324 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 874 \beta_{13} + 843 \beta_{12} + 1221 \beta_{11} + 110 \beta_{10} - 802 \beta_{9} - 161 \beta_{8} + \cdots + 5020 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1381 \beta_{13} - 709 \beta_{12} + 4736 \beta_{11} - 1118 \beta_{10} - 2448 \beta_{9} - 1468 \beta_{8} + \cdots + 21582 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6711 \beta_{13} + 6396 \beta_{12} + 9916 \beta_{11} + 756 \beta_{10} - 6422 \beta_{9} - 1521 \beta_{8} + \cdots + 36083 \) 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
2.67638
2.40446
2.28141
1.15628
1.10482
0.814716
0.0992940
0.0656838
−0.668880
−1.15093
−1.40019
−1.73208
−2.14277
−2.50820
−2.67638 0 5.16301 1.00000 0 −0.914080 −8.46540 0 −2.67638
1.2 −2.40446 0 3.78142 1.00000 0 −4.79843 −4.28335 0 −2.40446
1.3 −2.28141 0 3.20483 1.00000 0 −0.425177 −2.74872 0 −2.28141
1.4 −1.15628 0 −0.663011 1.00000 0 0.0469121 3.07919 0 −1.15628
1.5 −1.10482 0 −0.779371 1.00000 0 −2.51865 3.07071 0 −1.10482
1.6 −0.814716 0 −1.33624 1.00000 0 2.87567 2.71809 0 −0.814716
1.7 −0.0992940 0 −1.99014 1.00000 0 −0.183420 0.396197 0 −0.0992940
1.8 −0.0656838 0 −1.99569 1.00000 0 −4.02533 0.262452 0 −0.0656838
1.9 0.668880 0 −1.55260 1.00000 0 3.89603 −2.37626 0 0.668880
1.10 1.15093 0 −0.675359 1.00000 0 −4.95141 −3.07915 0 1.15093
1.11 1.40019 0 −0.0394812 1.00000 0 0.844557 −2.85565 0 1.40019
1.12 1.73208 0 1.00011 1.00000 0 1.33575 −1.73189 0 1.73208
1.13 2.14277 0 2.59145 1.00000 0 0.119004 1.26734 0 2.14277
1.14 2.50820 0 4.29107 1.00000 0 −3.30143 5.74646 0 2.50820
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(19\) \(1\)

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 9405.2.a.bq ✓ 14
3.b odd 2 1 9405.2.a.br yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9405.2.a.bq ✓ 14 1.a even 1 1 trivial
9405.2.a.br yes 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{14} + T_{2}^{13} - 19 T_{2}^{12} - 15 T_{2}^{11} + 137 T_{2}^{10} + 80 T_{2}^{9} - 467 T_{2}^{8} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{14} + 12 T_{7}^{13} + 18 T_{7}^{12} - 272 T_{7}^{11} - 974 T_{7}^{10} + 1228 T_{7}^{9} + 8260 T_{7}^{8} + \cdots + 4 \) Copy content Toggle raw display
\( T_{13}^{14} + 10 T_{13}^{13} - 40 T_{13}^{12} - 582 T_{13}^{11} - 11 T_{13}^{10} + 10416 T_{13}^{9} + \cdots + 43200 \) Copy content Toggle raw display
\( T_{17}^{14} + 16 T_{17}^{13} + 6 T_{17}^{12} - 942 T_{17}^{11} - 2935 T_{17}^{10} + 13598 T_{17}^{9} + \cdots - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T - 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 12 T^{13} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 10 T^{13} + \cdots + 43200 \) Copy content Toggle raw display
$17$ \( T^{14} + 16 T^{13} + \cdots - 64 \) Copy content Toggle raw display
$19$ \( (T + 1)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + 12 T^{13} + \cdots - 59743520 \) Copy content Toggle raw display
$29$ \( T^{14} - 231 T^{12} + \cdots - 53623040 \) Copy content Toggle raw display
$31$ \( T^{14} + 4 T^{13} + \cdots - 10548992 \) Copy content Toggle raw display
$37$ \( T^{14} + 14 T^{13} + \cdots - 6135952 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 1718932736 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 41586548404 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 530047375360 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 1858899968 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 3546831824 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 88156908800 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 788916944960 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 3029656592 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 127609978880 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 312081765760 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 3359191332544 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 7243418736 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 248856764272 \) Copy content Toggle raw display
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