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Properties

Label 928.2.k.e
Level $928$
Weight $2$
Character orbit 928.k
Analytic conductor $7.410$
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] = [928,2,Mod(191,928)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(928, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("928.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 928 = 2^{5} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 928.k (of order \(4\), degree \(2\), 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: \(7.41011730757\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 23x^{8} + 153x^{6} + 273x^{4} + 103x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} - \beta_{7} q^{5} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - 3 \beta_{8} - \beta_{7} + \cdots + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} - \beta_{7} q^{5} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - \beta_{9} - 8 \beta_{8} - \beta_{7} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{3} - 14 q^{11} + 14 q^{15} + 6 q^{17} - 12 q^{19} - 28 q^{21} + 2 q^{25} - 2 q^{27} - 28 q^{29} + 14 q^{31} + 4 q^{35} + 10 q^{37} + 6 q^{39} + 14 q^{41} - 30 q^{43} - 36 q^{45} + 6 q^{47} - 42 q^{49} - 56 q^{51} + 4 q^{53} - 42 q^{55} - 26 q^{61} + 32 q^{63} - 36 q^{65} - 56 q^{67} + 16 q^{69} - 36 q^{71} - 22 q^{73} + 8 q^{75} + 28 q^{77} - 6 q^{79} - 54 q^{81} - 16 q^{85} + 58 q^{87} + 58 q^{89} + 20 q^{91} + 52 q^{95} + 26 q^{97} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 23x^{8} + 153x^{6} + 273x^{4} + 103x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -15\nu^{8} - 335\nu^{6} - 2048\nu^{4} - 2351\nu^{2} + 685 ) / 284 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -21\nu^{8} - 469\nu^{6} - 2924\nu^{4} - 4257\nu^{2} - 1029 ) / 284 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15 \nu^{9} - 49 \nu^{8} - 335 \nu^{7} - 1118 \nu^{6} - 2048 \nu^{5} - 7296 \nu^{4} - 2351 \nu^{3} + \cdots - 2046 ) / 568 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15 \nu^{9} + 37 \nu^{8} + 335 \nu^{7} + 850 \nu^{6} + 2048 \nu^{5} + 5544 \nu^{4} + 2351 \nu^{3} + \cdots + 1458 ) / 568 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15 \nu^{9} - 37 \nu^{8} + 335 \nu^{7} - 850 \nu^{6} + 2048 \nu^{5} - 5544 \nu^{4} + 2351 \nu^{3} + \cdots - 1458 ) / 568 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15 \nu^{9} - 49 \nu^{8} + 335 \nu^{7} - 1118 \nu^{6} + 2048 \nu^{5} - 7296 \nu^{4} + 2351 \nu^{3} + \cdots - 2046 ) / 568 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\nu^{9} + 649\nu^{7} + 4372\nu^{5} + 7948\nu^{3} + 2437\nu ) / 284 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -49\nu^{9} - 1118\nu^{7} - 7296\nu^{5} - 12063\nu^{3} - 2614\nu ) / 426 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 388\nu^{9} + 8831\nu^{7} + 57216\nu^{5} + 91920\nu^{3} + 18799\nu ) / 852 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + 7\beta_{8} + 4\beta_{6} - 6\beta_{5} - 6\beta_{4} - 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{6} + 17\beta_{5} - 17\beta_{4} - 17\beta_{3} + 24\beta_{2} - 20\beta _1 + 100 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -58\beta_{9} - 218\beta_{8} - 10\beta_{7} - 89\beta_{6} + 133\beta_{5} + 133\beta_{4} + 89\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 116\beta_{6} - 128\beta_{5} + 128\beta_{4} + 116\beta_{3} - 128\beta_{2} + 116\beta _1 - 565 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 744\beta_{9} + 2952\beta_{8} + 256\beta_{7} + 1041\beta_{6} - 1505\beta_{5} - 1505\beta_{4} - 1041\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -3017\beta_{6} + 3553\beta_{5} - 3553\beta_{4} - 3017\beta_{3} + 2754\beta_{2} - 2802\beta _1 + 13242 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -4662\beta_{9} - 19179\beta_{8} - 2176\beta_{7} - 6208\beta_{6} + 8718\beta_{5} + 8718\beta_{4} + 6208\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/928\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(\beta_{8}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
3.51266i
0.356878i
− 2.85539i
− 1.42671i
− 0.587440i
− 3.51266i
− 0.356878i
2.85539i
1.42671i
0.587440i
0 −2.29821 − 2.29821i 0 2.13467i 0 − 1.90496i 0 7.56357i 0
191.2 0 −1.13340 − 1.13340i 0 − 1.87777i 0 − 4.61854i 0 − 0.430814i 0
191.3 0 −0.775651 − 0.775651i 0 2.46535i 0 2.99199i 0 − 1.79673i 0
191.4 0 1.01681 + 1.01681i 0 − 3.11240i 0 4.60823i 0 − 0.932203i 0
191.5 0 2.19045 + 2.19045i 0 0.390153i 0 0.923282i 0 6.59618i 0
447.1 0 −2.29821 + 2.29821i 0 − 2.13467i 0 1.90496i 0 − 7.56357i 0
447.2 0 −1.13340 + 1.13340i 0 1.87777i 0 4.61854i 0 0.430814i 0
447.3 0 −0.775651 + 0.775651i 0 − 2.46535i 0 − 2.99199i 0 1.79673i 0
447.4 0 1.01681 − 1.01681i 0 3.11240i 0 − 4.60823i 0 0.932203i 0
447.5 0 2.19045 − 2.19045i 0 − 0.390153i 0 − 0.923282i 0 − 6.59618i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
116.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 928.2.k.e ✓ 10
4.b odd 2 1 928.2.k.f yes 10
29.c odd 4 1 928.2.k.f yes 10
116.e even 4 1 inner 928.2.k.e ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.2.k.e ✓ 10 1.a even 1 1 trivial
928.2.k.e ✓ 10 116.e even 4 1 inner
928.2.k.f yes 10 4.b odd 2 1
928.2.k.f yes 10 29.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 2 T_{3}^{9} + 2 T_{3}^{8} - 2 T_{3}^{7} + 102 T_{3}^{6} + 186 T_{3}^{5} + 170 T_{3}^{4} + \cdots + 648 \) acting on \(S_{2}^{\mathrm{new}}(928, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 2 T^{9} + \cdots + 648 \) Copy content Toggle raw display
$5$ \( T^{10} + 24 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{10} + 56 T^{8} + \cdots + 12544 \) Copy content Toggle raw display
$11$ \( T^{10} + 14 T^{9} + \cdots + 223112 \) Copy content Toggle raw display
$13$ \( T^{10} + 64 T^{8} + \cdots + 9604 \) Copy content Toggle raw display
$17$ \( T^{10} - 6 T^{9} + \cdots + 1568 \) Copy content Toggle raw display
$19$ \( T^{10} + 12 T^{9} + \cdots + 839808 \) Copy content Toggle raw display
$23$ \( T^{10} + 100 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{10} + 28 T^{9} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( T^{10} - 14 T^{9} + \cdots + 28849608 \) Copy content Toggle raw display
$37$ \( T^{10} - 10 T^{9} + \cdots + 373248 \) Copy content Toggle raw display
$41$ \( T^{10} - 14 T^{9} + \cdots + 23328 \) Copy content Toggle raw display
$43$ \( T^{10} + 30 T^{9} + \cdots + 52488 \) Copy content Toggle raw display
$47$ \( T^{10} - 6 T^{9} + \cdots + 392 \) Copy content Toggle raw display
$53$ \( (T^{5} - 2 T^{4} + \cdots - 324)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 100 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{10} + 26 T^{9} + \cdots + 48649248 \) Copy content Toggle raw display
$67$ \( (T^{5} + 28 T^{4} + \cdots - 448)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 18 T^{4} + \cdots + 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 22 T^{9} + \cdots + 10179072 \) Copy content Toggle raw display
$79$ \( T^{10} + 6 T^{9} + \cdots + 20301192 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 337383424 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 2669174048 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 117596448 \) Copy content Toggle raw display
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