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Properties

Label 828.2.i.c
Level $828$
Weight $2$
Character orbit 828.i
Analytic conductor $6.612$
Analytic rank $0$
Dimension $12$
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] = [828,2,Mod(277,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.i (of order \(3\), 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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 10x^{9} - 14x^{8} - 6x^{7} + 51x^{6} - 18x^{5} - 126x^{4} + 270x^{3} - 486x + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} + \beta_1) q^{3} + (\beta_{11} - \beta_{8} + \beta_{6} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{9} + 2 \beta_{8} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} + \beta_1) q^{3} + (\beta_{11} - \beta_{8} + \beta_{6} + \cdots + 1) q^{5}+ \cdots + ( - 3 \beta_{11} + 2 \beta_{9} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 6 q^{5} + 10 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 6 q^{5} + 10 q^{7} + 4 q^{9} + 3 q^{11} - 19 q^{15} + 10 q^{19} + 6 q^{21} - 6 q^{23} - 4 q^{25} + 22 q^{27} + 7 q^{29} + q^{31} - 9 q^{33} - 38 q^{35} - 52 q^{37} + 39 q^{39} + 27 q^{41} - 9 q^{43} + 29 q^{45} + 16 q^{47} - 12 q^{49} - 34 q^{51} + 2 q^{53} - 4 q^{55} + 38 q^{57} - 16 q^{59} + 5 q^{61} + 27 q^{63} + q^{65} + 23 q^{67} - 2 q^{69} + 42 q^{71} + 46 q^{73} + 4 q^{75} + 5 q^{77} + 20 q^{79} - 8 q^{81} + 24 q^{83} + 34 q^{85} - 9 q^{87} - 18 q^{89} - 74 q^{91} + 8 q^{93} + 4 q^{95} - 11 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 10x^{9} - 14x^{8} - 6x^{7} + 51x^{6} - 18x^{5} - 126x^{4} + 270x^{3} - 486x + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{10} + 13 \nu^{9} - 6 \nu^{8} - 23 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} - 165 \nu^{4} + \cdots + 1053 ) / 486 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 3 \nu^{10} - 13 \nu^{9} + 16 \nu^{8} + 9 \nu^{7} - 52 \nu^{6} - 18 \nu^{5} + 147 \nu^{4} + \cdots - 1053 ) / 486 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 5 \nu^{10} + 15 \nu^{9} - 8 \nu^{8} - 17 \nu^{7} + 72 \nu^{6} + 24 \nu^{5} - 117 \nu^{4} + \cdots + 1215 ) / 486 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 17 \nu^{10} - 3 \nu^{9} - 37 \nu^{8} + 110 \nu^{7} + 66 \nu^{6} - 249 \nu^{5} + \cdots + 2187 ) / 1458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4 \nu^{11} + 20 \nu^{10} - 33 \nu^{9} + 5 \nu^{8} + 41 \nu^{7} - 72 \nu^{6} - 123 \nu^{5} + \cdots - 3159 ) / 1458 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 2\nu^{10} + 10\nu^{8} - 14\nu^{7} - 6\nu^{6} + 51\nu^{5} - 18\nu^{4} - 126\nu^{3} + 270\nu^{2} - 486 ) / 243 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{11} - 16 \nu^{10} + 3 \nu^{9} + 14 \nu^{8} - 22 \nu^{7} - 36 \nu^{6} + 120 \nu^{5} + \cdots + 1215 ) / 1458 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11 \nu^{11} - 37 \nu^{10} + 75 \nu^{9} - 34 \nu^{8} - 115 \nu^{7} + 189 \nu^{6} + 210 \nu^{5} + \cdots + 4860 ) / 1458 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13 \nu^{11} - 14 \nu^{10} + 30 \nu^{9} - 5 \nu^{8} - 8 \nu^{7} + 51 \nu^{6} + 51 \nu^{5} + \cdots + 5103 ) / 1458 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4 \nu^{11} - 11 \nu^{10} + 15 \nu^{9} - 5 \nu^{8} - 32 \nu^{7} + 27 \nu^{6} + 69 \nu^{5} - 144 \nu^{4} + \cdots + 972 ) / 486 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} - 2\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} - 2\beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{6} - 4 \beta_{5} + 3 \beta_{4} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{8} + 3\beta_{7} + \beta_{6} - 6\beta_{3} - 6\beta_{2} - 7\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{10} - 4\beta_{9} + 11\beta_{8} + 11\beta_{6} + \beta_{5} + 18\beta_{4} + 2\beta_{3} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5 \beta_{11} - 8 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 35 \beta_{6} + 16 \beta_{5} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 38 \beta_{11} + 6 \beta_{10} + 8 \beta_{9} + 8 \beta_{8} + 15 \beta_{7} - 28 \beta_{6} + 6 \beta_{5} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 78 \beta_{11} - 4 \beta_{10} + 94 \beta_{9} - 74 \beta_{8} - 92 \beta_{6} + 2 \beta_{5} - 54 \beta_{4} + \cdots - 19 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 16 \beta_{11} - 22 \beta_{10} + 98 \beta_{9} - 136 \beta_{8} + 60 \beta_{7} + 26 \beta_{6} + \cdots + 260 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 41 \beta_{11} + 60 \beta_{10} + 55 \beta_{9} + 55 \beta_{8} + 48 \beta_{7} + 235 \beta_{6} + 60 \beta_{5} + \cdots - 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
−1.35301 − 1.08137i
0.298220 − 1.70618i
−1.73189 + 0.0239614i
1.32849 − 1.11136i
1.61300 + 0.631060i
0.845191 + 1.51184i
−1.35301 + 1.08137i
0.298220 + 1.70618i
−1.73189 − 0.0239614i
1.32849 + 1.11136i
1.61300 − 0.631060i
0.845191 − 1.51184i
0 −1.61300 + 0.631060i 0 0.799696 + 1.38511i 0 1.87842 − 3.25352i 0 2.20353 − 2.03580i 0
277.2 0 −1.32849 − 1.11136i 0 −1.71977 − 2.97872i 0 1.47566 − 2.55591i 0 0.529765 + 2.95285i 0
277.3 0 −0.845191 + 1.51184i 0 0.461633 + 0.799571i 0 0.349475 − 0.605308i 0 −1.57130 − 2.55558i 0
277.4 0 −0.298220 − 1.70618i 0 −0.945890 − 1.63833i 0 −1.10236 + 1.90935i 0 −2.82213 + 1.01764i 0
277.5 0 1.35301 − 1.08137i 0 0.325104 + 0.563097i 0 −0.138406 + 0.239726i 0 0.661290 − 2.92621i 0
277.6 0 1.73189 + 0.0239614i 0 −1.92078 − 3.32688i 0 2.53722 − 4.39459i 0 2.99885 + 0.0829968i 0
553.1 0 −1.61300 − 0.631060i 0 0.799696 − 1.38511i 0 1.87842 + 3.25352i 0 2.20353 + 2.03580i 0
553.2 0 −1.32849 + 1.11136i 0 −1.71977 + 2.97872i 0 1.47566 + 2.55591i 0 0.529765 − 2.95285i 0
553.3 0 −0.845191 − 1.51184i 0 0.461633 − 0.799571i 0 0.349475 + 0.605308i 0 −1.57130 + 2.55558i 0
553.4 0 −0.298220 + 1.70618i 0 −0.945890 + 1.63833i 0 −1.10236 − 1.90935i 0 −2.82213 − 1.01764i 0
553.5 0 1.35301 + 1.08137i 0 0.325104 − 0.563097i 0 −0.138406 − 0.239726i 0 0.661290 + 2.92621i 0
553.6 0 1.73189 − 0.0239614i 0 −1.92078 + 3.32688i 0 2.53722 + 4.39459i 0 2.99885 − 0.0829968i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.i.c ✓ 12
3.b odd 2 1 2484.2.i.c 12
9.c even 3 1 inner 828.2.i.c ✓ 12
9.c even 3 1 7452.2.a.j 6
9.d odd 6 1 2484.2.i.c 12
9.d odd 6 1 7452.2.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
828.2.i.c ✓ 12 1.a even 1 1 trivial
828.2.i.c ✓ 12 9.c even 3 1 inner
2484.2.i.c 12 3.b odd 2 1
2484.2.i.c 12 9.d odd 6 1
7452.2.a.i 6 9.d odd 6 1
7452.2.a.j 6 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 6 T_{5}^{11} + 35 T_{5}^{10} + 72 T_{5}^{9} + 203 T_{5}^{8} + 67 T_{5}^{7} + 733 T_{5}^{6} + \cdots + 576 \) acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{12} - 10 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{12} + 38 T^{10} + \cdots + 2401 \) Copy content Toggle raw display
$17$ \( (T^{6} - 79 T^{4} + \cdots + 4056)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 5 T^{5} + \cdots - 808)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 1003432329 \) Copy content Toggle raw display
$31$ \( T^{12} - T^{11} + \cdots + 502681 \) Copy content Toggle raw display
$37$ \( (T^{6} + 26 T^{5} + \cdots - 33728)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 27 T^{11} + \cdots + 27468081 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 363436096 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 10086385761 \) Copy content Toggle raw display
$53$ \( (T^{6} - T^{5} + \cdots - 148056)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 275351018121 \) Copy content Toggle raw display
$61$ \( T^{12} - 5 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 384211543104 \) Copy content Toggle raw display
$71$ \( (T^{6} - 21 T^{5} + \cdots - 81)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 23 T^{5} + \cdots - 106281)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 20 T^{11} + \cdots + 90935296 \) Copy content Toggle raw display
$83$ \( T^{12} - 24 T^{11} + \cdots + 331776 \) Copy content Toggle raw display
$89$ \( (T^{6} + 9 T^{5} + \cdots - 8136)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 4929163264 \) Copy content Toggle raw display
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