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Properties

Label 9464.2.a.bm
Level $9464$
Weight $2$
Character orbit 9464.a
Self dual yes
Analytic conductor $75.570$
Analytic rank $0$
Dimension $12$
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] = [9464,2,Mod(1,9464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9464, 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("9464.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9464.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: \(75.5704204729\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 25 x^{10} + 52 x^{9} + 208 x^{8} - 464 x^{7} - 611 x^{6} + 1664 x^{5} + 119 x^{4} + \cdots - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 728)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{6} + 1) q^{5} + q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{6} + 1) q^{5} + q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{6} - \beta_{5}) q^{11} + (\beta_{11} - \beta_{9} - \beta_{7} + \cdots + \beta_1) q^{15}+ \cdots + ( - \beta_{11} - \beta_{10} - 2 \beta_{8} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} + 8 q^{5} + 12 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} + 8 q^{5} + 12 q^{7} + 18 q^{9} - 2 q^{11} - 2 q^{15} + 8 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} + 14 q^{25} - 10 q^{27} - 2 q^{29} - 16 q^{31} + 12 q^{33} + 8 q^{35} + 32 q^{37} - 22 q^{41} + 24 q^{43} + 42 q^{45} - 22 q^{47} + 12 q^{49} + 36 q^{51} - 24 q^{53} + 44 q^{55} + 28 q^{57} - 20 q^{59} + 30 q^{61} + 18 q^{63} + 22 q^{67} + 36 q^{69} - 12 q^{71} + 30 q^{73} + 22 q^{75} - 2 q^{77} + 10 q^{79} + 40 q^{81} - 12 q^{83} + 56 q^{85} + 20 q^{87} + 14 q^{89} + 20 q^{93} + 18 q^{95} - 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 25 x^{10} + 52 x^{9} + 208 x^{8} - 464 x^{7} - 611 x^{6} + 1664 x^{5} + 119 x^{4} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4939 \nu^{11} + 1444 \nu^{10} + 132955 \nu^{9} - 40938 \nu^{8} - 1263980 \nu^{7} + 397080 \nu^{6} + \cdots - 347444 ) / 229424 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4217 \nu^{11} + 4740 \nu^{10} + 107945 \nu^{9} - 118334 \nu^{8} - 947308 \nu^{7} + 997112 \nu^{6} + \cdots - 19756 ) / 114712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3228541 \nu^{11} - 4325356 \nu^{10} - 82888205 \nu^{9} + 112717510 \nu^{8} + 730062516 \nu^{7} + \cdots - 60383620 ) / 19042192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3475883 \nu^{11} - 5327444 \nu^{10} - 89499963 \nu^{9} + 139381338 \nu^{8} + 791497644 \nu^{7} + \cdots + 50293572 ) / 19042192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3656765 \nu^{11} - 5624988 \nu^{10} - 93837517 \nu^{9} + 147093926 \nu^{8} + 822575028 \nu^{7} + \cdots - 58156900 ) / 19042192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4035271 \nu^{11} + 5809044 \nu^{10} + 103726183 \nu^{9} - 150663442 \nu^{8} - 913601580 \nu^{7} + \cdots + 63910636 ) / 19042192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7739301 \nu^{11} - 11108476 \nu^{10} - 199960405 \nu^{9} + 289064102 \nu^{8} + 1777759860 \nu^{7} + \cdots + 119014492 ) / 19042192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 753219 \nu^{11} - 1154692 \nu^{10} - 19405715 \nu^{9} + 30150346 \nu^{8} + 171559772 \nu^{7} + \cdots + 20253956 ) / 1464784 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3255097 \nu^{11} - 4834088 \nu^{10} - 83790625 \nu^{9} + 126168002 \nu^{8} + 740084888 \nu^{7} + \cdots + 45968904 ) / 4760548 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{5} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 2\beta_{8} - \beta_{7} - 4\beta_{6} + \beta_{3} + 11\beta_{2} - \beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 16 \beta_{11} + 12 \beta_{10} + 3 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32 \beta_{11} - \beta_{10} - \beta_{9} + 26 \beta_{8} - 18 \beta_{7} - 62 \beta_{6} - \beta_{5} + \cdots + 328 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 201 \beta_{11} + 122 \beta_{10} + 58 \beta_{9} + 4 \beta_{8} + 55 \beta_{7} + 69 \beta_{6} + \cdots - 173 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 414 \beta_{11} - 19 \beta_{10} - 38 \beta_{9} + 278 \beta_{8} - 250 \beta_{7} - 762 \beta_{6} + \cdots + 3026 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2321 \beta_{11} + 1196 \beta_{10} + 799 \beta_{9} + 95 \beta_{8} + 753 \beta_{7} + 1084 \beta_{6} + \cdots - 2220 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5007 \beta_{11} - 296 \beta_{10} - 767 \beta_{9} + 2832 \beta_{8} - 3128 \beta_{7} - 8696 \beta_{6} + \cdots + 29061 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 25870 \beta_{11} + 11663 \beta_{10} + 9720 \beta_{9} + 1471 \beta_{8} + 9288 \beta_{7} + 14709 \beta_{6} + \cdots - 27340 \) 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
−3.29939
−2.82422
−2.03480
−1.75990
−0.0301869
0.447945
0.532720
1.01988
1.47043
2.43264
2.91186
3.13302
0 −3.29939 0 −0.585323 0 1.00000 0 7.88599 0
1.2 0 −2.82422 0 4.08885 0 1.00000 0 4.97621 0
1.3 0 −2.03480 0 2.24692 0 1.00000 0 1.14041 0
1.4 0 −1.75990 0 −0.457188 0 1.00000 0 0.0972504 0
1.5 0 −0.0301869 0 0.691852 0 1.00000 0 −2.99909 0
1.6 0 0.447945 0 0.513505 0 1.00000 0 −2.79934 0
1.7 0 0.532720 0 −3.61008 0 1.00000 0 −2.71621 0
1.8 0 1.01988 0 −2.02001 0 1.00000 0 −1.95984 0
1.9 0 1.47043 0 2.91465 0 1.00000 0 −0.837826 0
1.10 0 2.43264 0 3.98804 0 1.00000 0 2.91772 0
1.11 0 2.91186 0 −2.05353 0 1.00000 0 5.47892 0
1.12 0 3.13302 0 2.28231 0 1.00000 0 6.81582 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 9464.2.a.bm 12
13.b even 2 1 9464.2.a.bl 12
13.f odd 12 2 728.2.bm.c ✓ 24
52.l even 12 2 1456.2.cc.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.bm.c ✓ 24 13.f odd 12 2
1456.2.cc.g 24 52.l even 12 2
9464.2.a.bl 12 13.b even 2 1
9464.2.a.bm 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9464))\):

\( T_{3}^{12} - 2 T_{3}^{11} - 25 T_{3}^{10} + 52 T_{3}^{9} + 208 T_{3}^{8} - 464 T_{3}^{7} - 611 T_{3}^{6} + \cdots - 8 \) Copy content Toggle raw display
\( T_{5}^{12} - 8 T_{5}^{11} - 5 T_{5}^{10} + 172 T_{5}^{9} - 235 T_{5}^{8} - 1034 T_{5}^{7} + 2078 T_{5}^{6} + \cdots - 347 \) Copy content Toggle raw display
\( T_{11}^{12} + 2 T_{11}^{11} - 91 T_{11}^{10} - 134 T_{11}^{9} + 2847 T_{11}^{8} + 2130 T_{11}^{7} + \cdots - 25088 \) Copy content Toggle raw display
\( T_{17}^{12} - 8 T_{17}^{11} - 144 T_{17}^{10} + 1220 T_{17}^{9} + 7446 T_{17}^{8} - 68200 T_{17}^{7} + \cdots + 96651088 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{12} - 8 T^{11} + \cdots - 347 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots - 25088 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots + 96651088 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} + \cdots + 6016 \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} + \cdots - 815744 \) Copy content Toggle raw display
$29$ \( T^{12} + 2 T^{11} + \cdots + 12112 \) Copy content Toggle raw display
$31$ \( T^{12} + 16 T^{11} + \cdots - 13877888 \) Copy content Toggle raw display
$37$ \( T^{12} - 32 T^{11} + \cdots - 2163200 \) Copy content Toggle raw display
$41$ \( T^{12} + 22 T^{11} + \cdots - 31986176 \) Copy content Toggle raw display
$43$ \( T^{12} - 24 T^{11} + \cdots - 28126208 \) Copy content Toggle raw display
$47$ \( T^{12} + 22 T^{11} + \cdots + 197632 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 558967168 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 7494000208 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 960361664 \) Copy content Toggle raw display
$67$ \( T^{12} - 22 T^{11} + \cdots - 25539200 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 3243774976 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 14312045312 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1450283008 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 1123673792 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 232364032 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 8388995072 \) Copy content Toggle raw display
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