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Properties

Label 1296.4.c.e
Level $1296$
Weight $4$
Character orbit 1296.c
Analytic conductor $76.466$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,4,Mod(1295,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.1295");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 15 x^{8} + 26 x^{7} + 101 x^{6} + 396 x^{5} + 1292 x^{4} + 2864 x^{3} + 7860 x^{2} + \cdots + 26368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{9} + \beta_{3} + 2 \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{9} + \beta_{3} + 2 \beta_{2}) q^{7} + ( - \beta_{8} - \beta_{5} - \beta_{4} - 6) q^{11} + ( - 2 \beta_{8} - \beta_{5} - 11) q^{13} + ( - 2 \beta_{9} + \beta_{6} + \cdots - 12 \beta_{2}) q^{17}+ \cdots + (19 \beta_{8} + 10 \beta_{7} + \cdots - 134) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 54 q^{11} - 110 q^{13} + 306 q^{23} - 112 q^{25} + 990 q^{35} + 1100 q^{37} + 1206 q^{47} - 884 q^{49} + 846 q^{59} + 650 q^{61} + 5040 q^{71} + 736 q^{73} + 2646 q^{83} - 2904 q^{85} + 4644 q^{95} - 1106 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 15 x^{8} + 26 x^{7} + 101 x^{6} + 396 x^{5} + 1292 x^{4} + 2864 x^{3} + 7860 x^{2} + \cdots + 26368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{9} - 215 \nu^{8} + 2011 \nu^{7} - 6871 \nu^{6} + 17841 \nu^{5} - 21759 \nu^{4} + \cdots - 651008 ) / 52488 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29 \nu^{9} + 272 \nu^{8} - 487 \nu^{7} + 4348 \nu^{6} - 3117 \nu^{5} + 37158 \nu^{4} + \cdots + 454928 ) / 524880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 18 \nu^{8} + 23 \nu^{7} - 162 \nu^{6} - 507 \nu^{5} - 1212 \nu^{4} - 2596 \nu^{3} + \cdots - 9072 ) / 4860 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8 \nu^{9} + \nu^{8} + 112 \nu^{7} - 283 \nu^{6} + 156 \nu^{5} + 219 \nu^{4} - 15230 \nu^{3} + \cdots - 370208 ) / 17496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{9} - 29 \nu^{8} + 109 \nu^{7} - 217 \nu^{6} + 39 \nu^{5} + 1398 \nu^{4} - 1283 \nu^{3} + \cdots + 56344 ) / 2187 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 97 \nu^{9} + 196 \nu^{8} + 829 \nu^{7} - 5896 \nu^{6} + 18399 \nu^{5} - 11886 \nu^{4} + \cdots + 63184 ) / 87480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32 \nu^{9} + 23 \nu^{8} - 448 \nu^{7} - 677 \nu^{6} - 3540 \nu^{5} - 13107 \nu^{4} + \cdots - 326656 ) / 17496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4 \nu^{9} + 13 \nu^{8} - 56 \nu^{7} - 34 \nu^{6} - 78 \nu^{5} - 393 \nu^{4} - 728 \nu^{3} + \cdots - 575 ) / 2187 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 241 \nu^{9} - 646 \nu^{8} + 4751 \nu^{7} + 1702 \nu^{6} + 32037 \nu^{5} + 70368 \nu^{4} + \cdots + 2656400 ) / 104976 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} - 3\beta_{2} - \beta _1 + 4 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} - 3\beta_{8} - 2\beta_{7} + 4\beta_{6} - 2\beta_{4} + \beta_{3} - 10\beta_{2} - 2\beta _1 - 47 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + 6 \beta_{8} - 9 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} - 9 \beta_{4} + 21 \beta_{3} + \cdots - 294 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{9} + 45 \beta_{8} - 34 \beta_{7} - 34 \beta_{6} + 12 \beta_{5} - 10 \beta_{4} - \beta_{3} + \cdots - 703 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12 \beta_{9} + 63 \beta_{8} + 38 \beta_{7} - 162 \beta_{6} + 24 \beta_{5} - 34 \beta_{4} - 247 \beta_{3} + \cdots + 203 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 49 \beta_{9} - 165 \beta_{8} + 261 \beta_{7} - 158 \beta_{6} + 3 \beta_{5} + 21 \beta_{4} - 341 \beta_{3} + \cdots + 3891 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 203 \beta_{9} - 1587 \beta_{8} + 1255 \beta_{7} + 788 \beta_{6} - 159 \beta_{5} + 1159 \beta_{4} + \cdots + 37765 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 18 \beta_{9} - 2079 \beta_{8} - 1190 \beta_{7} + 6666 \beta_{6} - 1524 \beta_{5} + 4738 \beta_{4} + \cdots + 75829 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1537 \beta_{9} + 3396 \beta_{8} - 20007 \beta_{7} + 17294 \beta_{6} - 3879 \beta_{5} + 3729 \beta_{4} + \cdots - 203592 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1295.1
0.251202 − 2.90505i
2.22708 + 2.73757i
−1.92524 + 0.665541i
1.90731 − 2.85951i
−1.46035 + 1.71659i
−1.46035 − 1.71659i
1.90731 + 2.85951i
−1.92524 − 0.665541i
2.22708 − 2.73757i
0.251202 + 2.90505i
0 0 0 − 18.6154i 0 11.7916i 0 0 0
1295.2 0 0 0 − 15.3107i 0 − 4.87903i 0 0 0
1295.3 0 0 0 − 6.62728i 0 29.2925i 0 0 0
1295.4 0 0 0 − 6.15095i 0 1.97380i 0 0 0
1295.5 0 0 0 − 4.27738i 0 33.6483i 0 0 0
1295.6 0 0 0 4.27738i 0 − 33.6483i 0 0 0
1295.7 0 0 0 6.15095i 0 − 1.97380i 0 0 0
1295.8 0 0 0 6.62728i 0 − 29.2925i 0 0 0
1295.9 0 0 0 15.3107i 0 4.87903i 0 0 0
1295.10 0 0 0 18.6154i 0 − 11.7916i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1295.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.4.c.e 10
3.b odd 2 1 1296.4.c.f 10
4.b odd 2 1 1296.4.c.f 10
9.c even 3 1 144.4.s.c ✓ 10
9.c even 3 1 432.4.s.d 10
9.d odd 6 1 144.4.s.d yes 10
9.d odd 6 1 432.4.s.c 10
12.b even 2 1 inner 1296.4.c.e 10
36.f odd 6 1 144.4.s.d yes 10
36.f odd 6 1 432.4.s.c 10
36.h even 6 1 144.4.s.c ✓ 10
36.h even 6 1 432.4.s.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.4.s.c ✓ 10 9.c even 3 1
144.4.s.c ✓ 10 36.h even 6 1
144.4.s.d yes 10 9.d odd 6 1
144.4.s.d yes 10 36.f odd 6 1
432.4.s.c 10 9.d odd 6 1
432.4.s.c 10 36.f odd 6 1
432.4.s.d 10 9.c even 3 1
432.4.s.d 10 36.h even 6 1
1296.4.c.e 10 1.a even 1 1 trivial
1296.4.c.e 10 12.b even 2 1 inner
1296.4.c.f 10 3.b odd 2 1
1296.4.c.f 10 4.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 \(S_{4}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{10} + 681T_{5}^{8} + 142515T_{5}^{6} + 9992187T_{5}^{4} + 274155192T_{5}^{2} + 2469692592 \) Copy content Toggle raw display
\( T_{11}^{5} + 27T_{11}^{4} - 3600T_{11}^{3} - 86400T_{11}^{2} + 590247T_{11} + 4793661 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 2469692592 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 12527233200 \) Copy content Toggle raw display
$11$ \( (T^{5} + 27 T^{4} + \cdots + 4793661)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + 55 T^{4} + \cdots + 43356272)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 71\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 68\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( (T^{5} - 153 T^{4} + \cdots + 2895448032)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} - 550 T^{4} + \cdots - 2882353280)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 13\!\cdots\!03 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 46\!\cdots\!87 \) Copy content Toggle raw display
$47$ \( (T^{5} - 603 T^{4} + \cdots - 697070107152)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 1268547707205)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 325 T^{4} + \cdots - 135855377600)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 64\!\cdots\!83 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 77392428682176)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 54363390126368)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 46\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 133381694761488)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 68705085780451)^{2} \) Copy content Toggle raw display
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